{"id":51524,"date":"2025-08-11T00:04:36","date_gmt":"2025-08-10T16:04:36","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/51524.html"},"modified":"2025-08-11T00:04:36","modified_gmt":"2025-08-10T16:04:36","slug":"%e7%9f%a9%e9%98%b5%e5%88%86%e8%a7%a3%e6%8a%80%e6%9c%af%e8%a7%a3%e6%9e%90","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/51524.html","title":{"rendered":"\u77e9\u9635\u5206\u89e3\u6280\u672f\u89e3\u6790"},"content":{"rendered":"

\u77e9\u9635\u5206\u89e3\u6280\u672f\u89e3\u6790<\/h2>\n

\u77e9\u9635\u5206\u89e3\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u6700\u6838\u5fc3\u7684\u6280\u672f\u4e4b\u4e00，\u5b83\u5c06\u590d\u6742\u7684\u77e9\u9635\u8fd0\u7b97\u8f6c\u5316\u4e3a\u66f4\u7b80\u5355\u3001\u66f4\u6613\u5904\u7406\u7684\u5f62\u5f0f\u3002\u8fd9\u9879\u6280\u672f\u4e0d\u4ec5\u5728\u7406\u8bba\u6570\u5b66\u4e2d\u5360\u636e\u91cd\u8981\u5730\u4f4d，\u66f4\u662f\u73b0\u4ee3\u673a\u5668\u5b66\u4e60\u3001\u6570\u636e\u79d1\u5b66\u548c\u5de5\u7a0b\u8ba1\u7b97\u7684\u57fa\u7840\u5de5\u5177\u3002<\/p>\n

\u77e9\u9635\u5206\u89e3\u7684\u672c\u8d28\u4e0e\u610f\u4e49<\/h3>\n

\u77e9\u9635\u5206\u89e3\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u4e00\u4e2a\u590d\u6742\u7684\u77e9\u9635 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u8868\u793a\u4e3a\u82e5\u5e72\u4e2a\u5177\u6709\u7279\u6b8a\u6027\u8d28\u7684\u77e9\u9635\u7684\u4e58\u79ef\u3002\u8fd9\u79cd\u5206\u89e3\u63ed\u793a\u4e86\u539f\u59cb\u77e9\u9635\u7684\u5185\u5728\u7ed3\u6784，\u4f7f\u5f97\u8bb8\u591a\u770b\u4f3c\u56f0\u96be\u7684\u8ba1\u7b97\u53d8\u5f97\u7b80\u5355\u9ad8\u6548\u3002\u5c31\u50cf\u5c06\u4e00\u4e2a\u590d\u5408\u51fd\u6570\u5206\u89e3\u4e3a\u57fa\u672c\u51fd\u6570\u7684\u7ec4\u5408，\u77e9\u9635\u5206\u89e3\u4f7f\u6211\u4eec\u80fd\u591f\u901a\u8fc7\u7406\u89e3\u5404\u4e2a\u7ec4\u6210\u90e8\u5206\u6765\u7406\u89e3\u6574\u4f53\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d，\u77e9\u9635\u5206\u89e3\u7684\u4ef7\u503c\u4f53\u73b0\u5728\u591a\u4e2a\u65b9\u9762\u3002\u9996\u5148，\u5b83\u80fd\u591f\u5927\u5e45\u964d\u4f4e\u8ba1\u7b97\u590d\u6742\u5ea6\u2014\u2014\u4f8b\u5982\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax=b\\\\mathbf{Ax} = \\\\mathbf{b}<\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>，\u76f4\u63a5\u6c42\u89e3\u9700\u8981 O(n3)O(n^3)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u8ba1\u7b97\u91cf，\u800c\u901a\u8fc7 LU \u5206\u89e3\u540e\u53ea\u9700\u8981 O(n2)O(n^2)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u524d\u5411\u548c\u540e\u5411\u66ff\u6362\u3002\u5176\u6b21，\u77e9\u9635\u5206\u89e3\u80fd\u591f\u63d0\u9ad8\u6570\u503c\u7a33\u5b9a\u6027，\u907f\u514d\u76f4\u63a5\u8ba1\u7b97\u4e2d\u7684\u6570\u503c\u8bef\u5dee\u7d2f\u79ef\u3002\u6700\u91cd\u8981\u7684\u662f，\u4e0d\u540c\u7684\u5206\u89e3\u65b9\u6cd5\u63ed\u793a\u4e86\u77e9\u9635\u7684\u4e0d\u540c\u6027\u8d28，\u4e3a\u5404\u79cd\u5e94\u7528\u63d0\u4f9b\u4e86\u7406\u8bba\u57fa\u7840\u3002<\/p>\n

\u5947\u5f02\u503c\u5206\u89e3（SVD）：\u6700\u5f3a\u5927\u7684\u5206\u89e3\u5de5\u5177<\/h3>\n

\u5947\u5f02\u503c\u5206\u89e3\u53ef\u4ee5\u8bf4\u662f\u6240\u6709\u77e9\u9635\u5206\u89e3\u6280\u672f\u4e2d\u6700\u91cd\u8981\u3001\u6700\u901a\u7528\u7684\u4e00\u79cd\u3002\u5bf9\u4e8e\u4efb\u610f m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span>，\u65e0\u8bba\u662f\u65b9\u9635\u8fd8\u662f\u77e9\u5f62\u77e9\u9635，\u5b9e\u6570\u8fd8\u662f\u590d\u6570\u77e9\u9635，\u90fd\u5b58\u5728\u5947\u5f02\u503c\u5206\u89e3：<\/p>\n

A=U\u03a3VT\\\\mathbf{A} = \\\\mathbf{U\\\\Sigma V}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3V<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u662f m\u00d7mm \\\\times m<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span> \u7684\u6b63\u4ea4\u77e9\u9635，\u5305\u542b\u5de6\u5947\u5f02\u5411\u91cf；\u03a3\\\\mathbf{\\\\Sigma}<\/span><\/span>\u03a3<\/span><\/span><\/span><\/span><\/span> \u662f m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u7684\u5bf9\u89d2\u77e9\u9635，\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u79f0\u4e3a\u5947\u5f02\u503c，\u6309\u7167 \u03c31\u2265\u03c32\u2265\u2026\u2265\u03c3r\u22650\\\\sigma_1 \\\\geq \\\\sigma_2 \\\\geq \\\\ldots \\\\geq \\\\sigma_r \\\\geq 0<\/span><\/span>\u03c3<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u2026<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u7684\u987a\u5e8f\u6392\u5217；VT\\\\mathbf{V}^T<\/span><\/span>V<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f n\u00d7nn \\\\times n<\/span><\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u6b63\u4ea4\u77e9\u9635\u7684\u8f6c\u7f6e，\u5305\u542b\u53f3\u5947\u5f02\u5411\u91cf\u3002<\/p>\n

SVD \u7684\u51e0\u4f55\u76f4\u89c2<\/h4>\n

\u4ece\u51e0\u4f55\u89d2\u5ea6\u7406\u89e3，SVD \u5c06\u4efb\u4f55\u7ebf\u6027\u53d8\u6362\u5206\u89e3\u4e3a\u4e09\u4e2a\u57fa\u672c\u64cd\u4f5c\u7684\u7ec4\u5408\u3002\u60f3\u8c61\u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u5355\u4f4d\u5706，\u7ecf\u8fc7\u77e9\u9635 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u7684\u53d8\u6362\u540e\u4f1a\u53d8\u6210\u4e00\u4e2a\u692d\u5706\u3002SVD \u544a\u8bc9\u6211\u4eec\u8fd9\u4e2a\u53d8\u6362\u53ef\u4ee5\u5206\u89e3\u4e3a：\u9996\u5148\u901a\u8fc7 VT\\\\mathbf{V}^T<\/span><\/span>V<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8fdb\u884c\u65cb\u8f6c\u6216\u53cd\u5c04，\u5c06\u5750\u6807\u7cfb\u5bf9\u9f50\u5230\u7279\u5b9a\u65b9\u5411；\u7136\u540e\u901a\u8fc7 \u03a3\\\\mathbf{\\\\Sigma}<\/span><\/span>\u03a3<\/span><\/span><\/span><\/span><\/span> \u6cbf\u7740\u5750\u6807\u8f74\u8fdb\u884c\u62c9\u4f38\u6216\u538b\u7f29，\u62c9\u4f38\u7684\u6bd4\u4f8b\u5c31\u662f\u5947\u5f02\u503c；\u6700\u540e\u901a\u8fc7 U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u518d\u6b21\u65cb\u8f6c\u6216\u53cd\u5c04\u5230\u6700\u7ec8\u4f4d\u7f6e\u3002\u8fd9\u79cd\u51e0\u4f55\u7406\u89e3\u5bf9\u4e8e\u7406\u89e3\u4e3b\u6210\u5206\u5206\u6790（PCA）\u7279\u522b\u6709\u5e2e\u52a9\u3002\u5728\u9ad8\u7ef4\u6570\u636e\u7a7a\u95f4\u4e2d，SVD \u80fd\u591f\u627e\u5230\u6570\u636e\u53d8\u5316\u6700\u5927\u7684\u65b9\u5411（\u4e3b\u6210\u5206），\u8fd9\u4e9b\u65b9\u5411\u5c31\u662f\u53f3\u5947\u5f02\u5411\u91cf，\u800c\u5bf9\u5e94\u7684\u5947\u5f02\u503c\u8868\u793a\u4e86\u5728\u8be5\u65b9\u5411\u4e0a\u7684\u53d8\u5316\u7a0b\u5ea6\u3002<\/p>\n

\u8ba1\u7b97 SVD \u7684\u7b97\u6cd5<\/h4>\n

\u8ba1\u7b97 SVD \u7684\u6807\u51c6\u65b9\u6cd5\u662f\u5148\u8ba1\u7b97 ATA\\\\mathbf{A}^T\\\\mathbf{A}<\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u5f97\u5230 V\\\\mathbf{V}<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span>，\u518d\u8ba1\u7b97 AAT\\\\mathbf{AA}^T<\/span><\/span>AA<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u5f97\u5230 U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span>，\u5947\u5f02\u503c\u5219\u662f ATA\\\\mathbf{A}^T\\\\mathbf{A}<\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u7279\u5f81\u503c\u7684\u5e73\u65b9\u6839\u3002\u7136\u800c\u5728\u5b9e\u9645\u8ba1\u7b97\u4e2d，\u8fd9\u79cd\u65b9\u6cd5\u4f1a\u5bfc\u81f4\u6570\u503c\u7cbe\u5ea6\u635f\u5931，\u56e0\u4e3a ATA\\\\mathbf{A}^T\\\\mathbf{A}<\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u7684\u6761\u4ef6\u6570\u662f A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u6761\u4ef6\u6570\u7684\u5e73\u65b9\u3002\u73b0\u4ee3\u7b97\u6cd5\u91c7\u7528\u66f4\u7a33\u5b9a\u7684\u65b9\u6cd5，\u5982 Golub-Kahan \u53cc\u5bf9\u89d2\u5316\u7b97\u6cd5\u3002\u8be5\u7b97\u6cd5\u9996\u5148\u5c06\u77e9\u9635\u5316\u4e3a\u53cc\u5bf9\u89d2\u5f62\u5f0f，\u7136\u540e\u4f7f\u7528 QR \u8fed\u4ee3\u6216\u5206\u6cbb\u7b97\u6cd5\u8ba1\u7b97\u5947\u5f02\u503c\u3002\u6574\u4e2a\u8fc7\u7a0b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(min\u2061(mn2,m2n))O(\\\\min(mn^2, m^2n))<\/span><\/span>O<\/span>(<\/span>min<\/span>(<\/span>m<\/span>n<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>m<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>n<\/span>))<\/span><\/span><\/span><\/span><\/span>，\u5bf9\u4e8e\u65b9\u9635\u5c31\u662f O(n3)O(n^3)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u7279\u5f81\u503c\u5206\u89e3：\u63ed\u793a\u77e9\u9635\u7684\u672c\u8d28\u7279\u6027<\/h3>\n

\u7279\u5f81\u503c\u5206\u89e3\u662f\u7406\u89e3\u65b9\u9635\u6027\u8d28\u7684\u57fa\u672c\u5de5\u5177\u3002\u5bf9\u4e8e n\u00d7nn \\\\times n<\/span><\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u65b9\u9635 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span>，\u5982\u679c\u5b58\u5728 nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf，\u5219\u53ef\u4ee5\u5206\u89e3\u4e3a：<\/p>\n

A=V\u039bV\u22121\\\\mathbf{A} = \\\\mathbf{V\\\\Lambda V}^{-1}<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>V\u039bV<\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d V\\\\mathbf{V}<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span> \u7684\u5217\u662f\u7279\u5f81\u5411\u91cf，\u039b\\\\mathbf{\\\\Lambda}<\/span><\/span>\u039b<\/span><\/span><\/span><\/span><\/span> \u662f\u5bf9\u89d2\u77e9\u9635，\u5bf9\u89d2\u7ebf\u5143\u7d20\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u3002\u5bf9\u4e8e\u5b9e\u5bf9\u79f0\u77e9\u9635，\u7279\u5f81\u5411\u91cf\u662f\u6b63\u4ea4\u7684，\u5206\u89e3\u7b80\u5316\u4e3a A=Q\u039bQT\\\\mathbf{A} = \\\\mathbf{Q\\\\Lambda Q}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>Q\u039bQ<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u7279\u5f81\u503c\u7684\u7269\u7406\u610f\u4e49<\/h4>\n

\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u6709\u7740\u6df1\u523b\u7684\u7269\u7406\u610f\u4e49\u3002\u7279\u5f81\u5411\u91cf\u4ee3\u8868\u4e86\u53d8\u6362\u7684\u4e0d\u53d8\u65b9\u5411\u2014\u2014\u6cbf\u7740\u7279\u5f81\u5411\u91cf\u65b9\u5411\u7684\u5411\u91cf\u5728\u53d8\u6362\u540e\u4ecd\u7136\u4fdd\u6301\u540c\u4e00\u65b9\u5411，\u53ea\u662f\u957f\u5ea6\u53d1\u751f\u4e86\u53d8\u5316，\u53d8\u5316\u7684\u500d\u6570\u5c31\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u3002\u5728\u632f\u52a8\u5206\u6790\u4e2d，\u7279\u5f81\u5411\u91cf\u5bf9\u5e94\u632f\u52a8\u6a21\u6001，\u7279\u5f81\u503c\u5bf9\u5e94\u632f\u52a8\u9891\u7387\u7684\u5e73\u65b9；\u5728\u91cf\u5b50\u529b\u5b66\u4e2d，\u7279\u5f81\u5411\u91cf\u662f\u91cf\u5b50\u6001，\u7279\u5f81\u503c\u662f\u53ef\u89c2\u6d4b\u91cf\u7684\u6d4b\u91cf\u503c\u3002\u5176\u4e2d\u77e9\u9635\u7684\u8bb8\u591a\u91cd\u8981\u6027\u8d28\u90fd\u53ef\u4ee5\u901a\u8fc7\u7279\u5f81\u503c\u6765\u8868\u8fbe\u3002\u77e9\u9635\u7684\u8ff9（\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e4b\u548c）\u7b49\u4e8e\u6240\u6709\u7279\u5f81\u503c\u4e4b\u548c：tr(A)=\u2211i\u03bbi\\\\text{tr}(\\\\mathbf{A}) = \\\\sum_i \\\\lambda_i<\/span><\/span>tr<\/span><\/span>(<\/span>A<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2211<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>；\u884c\u5217\u5f0f\u7b49\u4e8e\u6240\u6709\u7279\u5f81\u503c\u4e4b\u79ef：det\u2061(A)=\u220fi\u03bbi\\\\det(\\\\mathbf{A}) = \\\\prod_i \\\\lambda_i<\/span><\/span>det<\/span>(<\/span>A<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u220f<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>；\u77e9\u9635\u7684\u79e9\u7b49\u4e8e\u975e\u96f6\u7279\u5f81\u503c\u7684\u4e2a\u6570\u3002\u8fd9\u4e9b\u5173\u7cfb\u5728\u7406\u8bba\u5206\u6790\u548c\u5b9e\u9645\u8ba1\u7b97\u4e2d\u90fd\u975e\u5e38\u6709\u7528\u3002<\/p>\n

\u8ba1\u7b97\u7279\u5f81\u503c\u7684\u6570\u503c\u65b9\u6cd5<\/h4>\n

\u7406\u8bba\u4e0a，\u7279\u5f81\u503c\u662f\u7279\u5f81\u591a\u9879\u5f0f det\u2061(A\u2212\u03bbI)=0\\\\det(\\\\mathbf{A} – \\\\lambda\\\\mathbf{I}) = 0<\/span><\/span>det<\/span>(<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03bb<\/span>I<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u7684\u6839，\u4f46\u76f4\u63a5\u6c42\u89e3\u9ad8\u6b21\u65b9\u7a0b\u5728\u6570\u503c\u4e0a\u662f\u4e0d\u7a33\u5b9a\u7684\u3002\u5b9e\u9645\u8ba1\u7b97\u4e2d\u4f7f\u7528\u8fed\u4ee3\u65b9\u6cd5，\u5176\u4e2d\u6700\u91cd\u8981\u7684\u662f QR \u7b97\u6cd5\u3002QR \u7b97\u6cd5\u7684\u57fa\u672c\u601d\u60f3\u662f\u901a\u8fc7\u76f8\u4f3c\u53d8\u6362\u5c06\u77e9\u9635\u5316\u4e3a\u4e0a\u4e09\u89d2\u5f62\u5f0f（Schur \u5f62\u5f0f），\u5bf9\u89d2\u7ebf\u5143\u7d20\u5c31\u662f\u7279\u5f81\u503c\u3002\u7b97\u6cd5\u6b65\u9aa4\u662f：\u5c06 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u8fdb\u884c QR \u5206\u89e3\u5f97\u5230 A=Q1R1\\\\mathbf{A} = \\\\mathbf{Q}_1\\\\mathbf{R}_1<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u7136\u540e\u8ba1\u7b97 A1=R1Q1\\\\mathbf{A}_1 = \\\\mathbf{R}_1\\\\mathbf{Q}_1<\/span><\/span>A<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>R<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u91cd\u590d\u8fd9\u4e2a\u8fc7\u7a0b\u3002\u5728\u9002\u5f53\u7684\u6761\u4ef6\u4e0b，\u5e8f\u5217 {Ak}\\\\{\\\\mathbf{A}_k\\\\}<\/span><\/span>{<\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u6536\u655b\u5230\u4e0a\u4e09\u89d2\u77e9\u9635\u3002\u4e3a\u4e86\u52a0\u901f\u6536\u655b，\u5b9e\u9645\u5b9e\u73b0\u4e2d\u4f1a\u5148\u5c06\u77e9\u9635\u5316\u4e3a Hessenberg \u5f62\u5f0f，\u5e76\u4f7f\u7528\u4f4d\u79fb\u7b56\u7565\u3002\u5bf9\u4e8e\u5927\u89c4\u6a21\u7a00\u758f\u77e9\u9635，Lanczos \u7b97\u6cd5\u548c Arnoldi \u7b97\u6cd5\u66f4\u52a0\u9ad8\u6548\u3002\u8fd9\u4e9b\u65b9\u6cd5\u57fa\u4e8e Krylov \u5b50\u7a7a\u95f4，\u53ea\u8ba1\u7b97\u90e8\u5206\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf，\u7279\u522b\u9002\u5408\u53ea\u9700\u8981\u6700\u5927\u6216\u6700\u5c0f\u51e0\u4e2a\u7279\u5f81\u503c\u7684\u60c5\u51b5\u3002<\/p>\n

LU \u5206\u89e3：\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u4e3b\u529b<\/h3>\n

LU \u5206\u89e3\u5c06\u65b9\u9635\u5206\u89e3\u4e3a\u4e0b\u4e09\u89d2\u77e9\u9635 L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u548c\u4e0a\u4e09\u89d2\u77e9\u9635 U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u7684\u4e58\u79ef：<\/p>\n

PA=LU\\\\mathbf{PA} = \\\\mathbf{LU}<\/span><\/span>PA<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>LU<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d P\\\\mathbf{P}<\/span><\/span>P<\/span><\/span><\/span><\/span><\/span> \u662f\u7f6e\u6362\u77e9\u9635，\u7528\u4e8e\u6570\u503c\u7a33\u5b9a\u6027\u3002LU \u5206\u89e3\u672c\u8d28\u4e0a\u662f\u9ad8\u65af\u6d88\u5143\u6cd5\u7684\u77e9\u9635\u5f62\u5f0f，L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u8bb0\u5f55\u4e86\u6d88\u5143\u8fc7\u7a0b\u4e2d\u7684\u4e58\u6570，U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u662f\u6d88\u5143\u540e\u7684\u4e0a\u4e09\u89d2\u77e9\u9635\u3002<\/p>\n

LU \u5206\u89e3\u7684\u8ba1\u7b97\u8fc7\u7a0b<\/h4>\n

Doolittle \u7b97\u6cd5\u662f\u8ba1\u7b97 LU \u5206\u89e3\u7684\u6807\u51c6\u65b9\u6cd5\u3002\u7b97\u6cd5\u9010\u5217\u5904\u7406\u77e9\u9635，\u5bf9\u4e8e\u7b2c kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u5217，\u9996\u5148\u9009\u62e9\u4e3b\u5143（\u90e8\u5206\u9009\u4e3b\u5143\u7b56\u7565\u9009\u62e9\u8be5\u5217\u7edd\u5bf9\u503c\u6700\u5927\u7684\u5143\u7d20），\u7136\u540e\u8ba1\u7b97 L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u7b2c kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u5217\u548c U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u7684\u7b2c kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u884c\u3002\u5177\u4f53\u8ba1\u7b97\u516c\u5f0f\u4e3a：<\/p>\n

\u5bf9\u4e8e U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u7684\u5143\u7d20：
\nuij=aij\u2212\u2211k=1i\u22121likukju_{ij} = a_{ij} – \\\\sum_{k=1}^{i-1} l_{ik}u_{kj}<\/span><\/span>u<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>i<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>kj<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u4e8e L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u7684\u5143\u7d20：
\nlij=aij\u2212\u2211k=1j\u22121likukjujjl_{ij} = \\\\frac{a_{ij} – \\\\sum_{k=1}^{j-1} l_{ik}u_{kj}}{u_{jj}}<\/span><\/span>l<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>jj<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u2211<\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>j<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>kj<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6574\u4e2a\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(2n3\/3)O(2n^3\/3)<\/span><\/span>O<\/span>(<\/span>2<\/span>n<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/3<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u7ea6\u4e3a\u77e9\u9635\u4e58\u6cd5\u7684 2\/32\/3<\/span><\/span>2\/3<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u6570\u503c\u7a33\u5b9a\u6027\u4e0e\u9009\u4e3b\u5143\u7b56\u7565<\/h4>\n

\u4e0d\u4f7f\u7528\u9009\u4e3b\u5143\u7684 LU \u5206\u89e3\u53ef\u80fd\u5728\u6570\u503c\u4e0a\u4e0d\u7a33\u5b9a\u3002\u8003\u8651\u77e9\u9635：
\n[\u03f5111]\\\\begin{bmatrix} \\\\epsilon & 1 \\\\\\\\ 1 & 1 \\\\end{bmatrix}<\/span><\/span>[<\/span><\/span><\/span>\u03f5<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5f53 \u03f5\\\\epsilon<\/span><\/span>\u03f5<\/span><\/span><\/span><\/span><\/span> \u5f88\u5c0f\u65f6，\u76f4\u63a5 LU \u5206\u89e3\u4f1a\u4ea7\u751f\u5f88\u5927\u7684\u6570\u503c\u8bef\u5dee\u3002\u90e8\u5206\u9009\u4e3b\u5143\u901a\u8fc7\u4ea4\u6362\u884c\u6765\u786e\u4fdd \u2223lij\u2223\u22641|l_{ij}| \\\\leq 1<\/span><\/span>\u2223<\/span>l<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，\u5927\u5927\u63d0\u9ad8\u4e86\u6570\u503c\u7a33\u5b9a\u6027\u3002\u7406\u8bba\u4e0a，\u589e\u957f\u56e0\u5b50（\u8861\u91cf\u6570\u503c\u7a33\u5b9a\u6027\u7684\u6307\u6807）\u6700\u574f\u60c5\u51b5\u4e0b\u53ef\u8fbe 2n\u221212^{n-1}<\/span><\/span>2<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4f46\u5b9e\u9645\u95ee\u9898\u4e2d\u5f88\u5c11\u51fa\u73b0\u5982\u6b64\u6781\u7aef\u7684\u60c5\u51b5\u3002<\/p>\n

\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/h4>\n

\u6709\u4e86 LU \u5206\u89e3，\u6c42\u89e3 Ax=b\\\\mathbf{Ax} = \\\\mathbf{b}<\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u53d8\u6210\u4e24\u6b65：\u9996\u5148\u6c42\u89e3 Ly=Pb\\\\mathbf{Ly} = \\\\mathbf{Pb}<\/span><\/span>Ly<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Pb<\/span><\/span><\/span><\/span><\/span><\/span>（\u524d\u5411\u66ff\u6362），\u7136\u540e\u6c42\u89e3 Ux=y\\\\mathbf{Ux} = \\\\mathbf{y}<\/span><\/span>Ux<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>y<\/span><\/span><\/span><\/span><\/span>（\u540e\u5411\u66ff\u6362）\u3002\u6bcf\u6b65\u53ea\u9700\u8981 O(n2)O(n^2)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u7684\u8fd0\u7b97，\u8fdc\u6bd4\u76f4\u63a5\u6c42\u9006\u77e9\u9635\u9ad8\u6548\u3002\u5f53\u9700\u8981\u6c42\u89e3\u591a\u4e2a\u53f3\u7aef\u5411\u91cf\u65f6，\u53ea\u9700\u8fdb\u884c\u4e00\u6b21 LU \u5206\u89e3，\u7136\u540e\u5bf9\u6bcf\u4e2a\u53f3\u7aef\u5411\u91cf\u8fdb\u884c\u524d\u5411\u548c\u540e\u5411\u66ff\u6362，\u8fd9\u5728\u8bb8\u591a\u5de5\u7a0b\u5e94\u7528\u4e2d\u975e\u5e38\u6709\u7528\u3002<\/p>\n

QR \u5206\u89e3：\u6570\u503c\u7a33\u5b9a\u6027\u7684\u5178\u8303<\/h3>\n

QR \u5206\u89e3\u5c06\u77e9\u9635\u5206\u89e3\u4e3a\u6b63\u4ea4\u77e9\u9635 Q\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> \u548c\u4e0a\u4e09\u89d2\u77e9\u9635 R\\\\mathbf{R}<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span> \u7684\u4e58\u79ef：<\/p>\n

A=QR\\\\mathbf{A} = \\\\mathbf{QR}<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>QR<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d QTQ=I\\\\mathbf{Q}^T\\\\mathbf{Q} = \\\\mathbf{I}<\/span><\/span>Q<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span>，\u4fdd\u8bc1\u4e86\u826f\u597d\u7684\u6570\u503c\u6027\u8d28\u3002QR \u5206\u89e3\u5728\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u3001\u7279\u5f81\u503c\u8ba1\u7b97\u7b49\u9886\u57df\u6709\u7740\u91cd\u8981\u5e94\u7528\u3002<\/p>\n

Gram-Schmidt \u6b63\u4ea4\u5316\u4e0e\u6570\u503c\u7a33\u5b9a\u6027<\/h4>\n

\u7ecf\u5178\u7684 Gram-Schmidt \u8fc7\u7a0b\u901a\u8fc7\u9010\u4e2a\u6b63\u4ea4\u5316\u5217\u5411\u91cf\u6765\u6784\u9020 Q\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span>：\u5bf9\u4e8e\u7b2c kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u4e2a\u5217\u5411\u91cf，\u51cf\u53bb\u5b83\u5728\u524d k\u22121k-1<\/span><\/span>k<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u4e2a\u6b63\u4ea4\u5411\u91cf\u4e0a\u7684\u6295\u5f71，\u7136\u540e\u5f52\u4e00\u5316\u3002\u7136\u800c，\u7531\u4e8e\u820d\u5165\u8bef\u5dee\u7684\u7d2f\u79ef，\u7ecf\u5178 Gram-Schmidt \u5728\u6570\u503c\u4e0a\u662f\u4e0d\u7a33\u5b9a\u7684\u3002\u6539\u8fdb\u7684 Gram-Schmidt \u7b97\u6cd5\u901a\u8fc7\u6539\u53d8\u8ba1\u7b97\u987a\u5e8f\u63d0\u9ad8\u4e86\u7a33\u5b9a\u6027，\u4f46\u6700\u7a33\u5b9a\u7684\u65b9\u6cd5\u662f Householder \u53d8\u6362\u3002Householder \u53d8\u6362\u4f7f\u7528\u53cd\u5c04\u77e9\u9635 H=I\u22122vvT\\\\mathbf{H} = \\\\mathbf{I} – 2\\\\mathbf{vv}^T<\/span><\/span>H<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span>vv<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6765\u6d88\u53bb\u5411\u91cf\u7684\u7279\u5b9a\u5206\u91cf\u3002\u6bcf\u6b21\u53d8\u6362\u5c06\u4e00\u5217\u7684\u5b50\u5bf9\u89d2\u5143\u7d20\u5316\u4e3a\u96f6，n\u22121n-1<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u6b21\u53d8\u6362\u540e\u5f97\u5230\u4e0a\u4e09\u89d2\u77e9\u9635 R\\\\mathbf{R}<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

QR \u5206\u89e3\u5728\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u4e2d\u7684\u5e94\u7528<\/h4>\n

\u5bf9\u4e8e\u8d85\u5b9a\u65b9\u7a0b\u7ec4 Ax\u2248b\\\\mathbf{Ax} \\\\approx \\\\mathbf{b}<\/span><\/span>Ax<\/span><\/span><\/span>\u2248<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>（m>nm > n<\/span><\/span>m<\/span><\/span>><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>），\u6700\u5c0f\u4e8c\u4e58\u89e3\u6ee1\u8db3\u6b63\u89c4\u65b9\u7a0b ATAx=ATb\\\\mathbf{A}^T\\\\mathbf{Ax} = \\\\mathbf{A}^T\\\\mathbf{b}<\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>\u3002\u76f4\u63a5\u6c42\u89e3\u6b63\u89c4\u65b9\u7a0b\u4f1a\u4f7f\u6761\u4ef6\u6570\u5e73\u65b9，\u964d\u4f4e\u6570\u503c\u7a33\u5b9a\u6027\u3002\u4f7f\u7528 QR \u5206\u89e3，\u95ee\u9898\u8f6c\u5316\u4e3a：<\/p>\n

\u2225Ax\u2212b\u22252=\u2225QRx\u2212b\u22252=\u2225Rx\u2212QTb\u22252\\\\|\\\\mathbf{Ax} – \\\\mathbf{b}\\\\|^2 = \\\\|\\\\mathbf{QRx} – \\\\mathbf{b}\\\\|^2 = \\\\|\\\\mathbf{Rx} – \\\\mathbf{Q}^T\\\\mathbf{b}\\\\|^2<\/span><\/span>\u2225<\/span>Ax<\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>QRx<\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>Rx<\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>Q<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531\u4e8e Q\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u4ea4\u77e9\u9635，\u4e0d\u6539\u53d8\u5411\u91cf\u7684\u8303\u6570\u3002\u56e0\u6b64\u53ea\u9700\u6c42\u89e3\u4e0a\u4e09\u89d2\u65b9\u7a0b\u7ec4 Rx=QTb\\\\mathbf{Rx} = \\\\mathbf{Q}^T\\\\mathbf{b}<\/span><\/span>Rx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u7684\u524d nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u4e2a\u65b9\u7a0b\u5373\u53ef\u3002\u8fd9\u79cd\u65b9\u6cd5\u907f\u514d\u4e86\u5f62\u6210 ATA\\\\mathbf{A}^T\\\\mathbf{A}<\/span><\/span>A<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span>，\u4fdd\u6301\u4e86\u539f\u95ee\u9898\u7684\u6761\u4ef6\u6570，\u5728\u6570\u503c\u4e0a\u66f4\u52a0\u7a33\u5b9a\u3002<\/p>\n

QR \u7b97\u6cd5\u8ba1\u7b97\u7279\u5f81\u503c<\/h4>\n

QR \u7b97\u6cd5\u662f\u8ba1\u7b97\u77e9\u9635\u5168\u90e8\u7279\u5f81\u503c\u7684\u6807\u51c6\u65b9\u6cd5\u3002\u57fa\u672c\u601d\u60f3\u662f\u901a\u8fc7\u4e00\u7cfb\u5217\u76f8\u4f3c\u53d8\u6362\u5c06\u77e9\u9635\u5316\u4e3a\u4e0a\u4e09\u89d2\u5f62\u5f0f\u3002\u7b97\u6cd5\u7684\u6bcf\u4e00\u6b65\u8fdb\u884c QR \u5206\u89e3 Ak=QkRk\\\\mathbf{A}_k = \\\\mathbf{Q}_k\\\\mathbf{R}_k<\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u7136\u540e\u5f62\u6210 Ak+1=RkQk\\\\mathbf{A}_{k+1} = \\\\mathbf{R}_k\\\\mathbf{Q}_k<\/span><\/span>A<\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>R<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u7531\u4e8e Ak+1=QkTAkQk\\\\mathbf{A}_{k+1} = \\\\mathbf{Q}_k^T\\\\mathbf{A}_k\\\\mathbf{Q}_k<\/span><\/span>A<\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u4fdd\u6301\u4e86\u7279\u5f81\u503c\u4e0d\u53d8\u3002\u5728\u9002\u5f53\u6761\u4ef6\u4e0b，{Ak}\\\\{\\\\mathbf{A}_k\\\\}<\/span><\/span>{<\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u6536\u655b\u5230\u4e0a\u4e09\u89d2\u77e9\u9635（\u5b9e Schur \u5f62\u5f0f），\u5bf9\u89d2\u7ebf\u5143\u7d20\u5c31\u662f\u7279\u5f81\u503c\u3002\u5b9e\u9645\u5b9e\u73b0\u4e2d，\u5148\u5c06\u77e9\u9635\u5316\u4e3a Hessenberg \u5f62\u5f0f\u53ef\u4ee5\u5927\u5927\u51cf\u5c11\u8ba1\u7b97\u91cf，\u6bcf\u6b65 QR \u5206\u89e3\u53ea\u9700\u8981 O(n2)O(n^2)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u800c\u4e0d\u662f O(n3)O(n^3)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

Cholesky \u5206\u89e3：\u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635\u7684\u4e13\u5c5e\u65b9\u6cd5<\/h3>\n

\u5bf9\u4e8e\u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635，Cholesky \u5206\u89e3\u63d0\u4f9b\u4e86\u6700\u9ad8\u6548\u7684\u5206\u89e3\u65b9\u6cd5：<\/p>\n

A=LLT\\\\mathbf{A} = \\\\mathbf{LL}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>LL<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u662f\u4e0b\u4e09\u89d2\u77e9\u9635，\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e3a\u6b63\u3002Cholesky \u5206\u89e3\u53ef\u4ee5\u770b\u4f5c\u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635\u7684"\u5e73\u65b9\u6839"\u3002<\/p>\n

\u8ba1\u7b97 Cholesky \u5206\u89e3<\/h4>\n

Cholesky \u5206\u89e3\u7684\u8ba1\u7b97\u76f8\u5bf9\u7b80\u5355\u3002\u5bf9\u89d2\u7ebf\u5143\u7d20\u901a\u8fc7\u4e0b\u5f0f\u8ba1\u7b97，\u786e\u4fdd\u4e86\u5b9e\u6570\u548c\u6b63\u503c：
\nlii=aii\u2212\u2211k=1i\u22121lik2l_{ii} = \\\\sqrt{a_{ii} – \\\\sum_{k=1}^{i-1} l_{ik}^2}<\/span><\/span>l<\/span><\/span>ii<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>ii<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>i<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u975e\u5bf9\u89d2\u7ebf\u5143\u7d20\u901a\u8fc7\u4e0b\u5f0f\u8ba1\u7b97：
\nlij=aij\u2212\u2211k=1j\u22121likljkljjl_{ij} = \\\\frac{a_{ij} – \\\\sum_{k=1}^{j-1} l_{ik}l_{jk}}{l_{jj}}<\/span><\/span>l<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>jj<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u2211<\/span><\/span>k<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>j<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>ik<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>jk<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7b97\u6cd5\u53ea\u9700\u8981\u5904\u7406\u4e0b\u4e09\u89d2\u90e8\u5206，\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a O(n3\/3)O(n^3\/3)<\/span><\/span>O<\/span>(<\/span>n<\/span><\/span>3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/3<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u7ea6\u4e3a LU \u5206\u89e3\u7684\u4e00\u534a\u3002\u7a7a\u95f4\u4e0a\u4e5f\u53ea\u9700\u8981\u5b58\u50a8\u4e0b\u4e09\u89d2\u90e8\u5206，\u8282\u7701\u4e86\u4e00\u534a\u7684\u5185\u5b58\u3002<\/p>\n

\u6b63\u5b9a\u6027\u7684\u51e0\u4f55\u610f\u4e49<\/h4>\n

\u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635\u5b9a\u4e49\u4e86\u4e00\u4e2a\u692d\u7403\u4f53：xTAx=1\\\\mathbf{x}^T\\\\mathbf{Ax} = 1<\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002Cholesky \u5206\u89e3 A=LLT\\\\mathbf{A} = \\\\mathbf{LL}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>LL<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5c06\u8fd9\u4e2a\u692d\u7403\u4f53\u7684\u63cf\u8ff0\u5206\u89e3\u4e3a\u4e24\u6b65\u53d8\u6362：\u9996\u5148\u901a\u8fc7 LT\\\\mathbf{L}^T<\/span><\/span>L<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5c06\u6807\u51c6\u7403\u9762\u53d8\u6362\u5230\u65b0\u7684\u5750\u6807\u7cfb，\u7136\u540e\u901a\u8fc7 L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u8fdb\u884c\u76f8\u540c\u7684\u53d8\u6362\u3002\u8fd9\u4fdd\u8bc1\u4e86\u692d\u7403\u4f53\u7684\u6240\u6709\u4e3b\u8f74\u957f\u5ea6\u90fd\u662f\u6b63\u7684，\u5bf9\u5e94\u4e8e\u6b63\u5b9a\u6027\u7684\u8981\u6c42\u3002\u5728\u6982\u7387\u8bba\u4e2d，\u591a\u5143\u6b63\u6001\u5206\u5e03\u7684\u534f\u65b9\u5dee\u77e9\u9635\u603b\u662f\u5bf9\u79f0\u6b63\u5b9a\u7684\u3002Cholesky \u5206\u89e3\u5728\u751f\u6210\u591a\u5143\u6b63\u6001\u968f\u673a\u6570\u65f6\u7279\u522b\u6709\u7528：\u5982\u679c z\\\\mathbf{z}<\/span><\/span>z<\/span><\/span><\/span><\/span><\/span> \u662f\u6807\u51c6\u6b63\u6001\u968f\u673a\u5411\u91cf，\u90a3\u4e48 x=Lz+\u03bc\\\\mathbf{x} = \\\\mathbf{Lz} + \\\\boldsymbol{\\\\mu}<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>Lz<\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03bc<\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u670d\u4ece\u5747\u503c\u4e3a \u03bc\\\\boldsymbol{\\\\mu}<\/span><\/span>\u03bc<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3001\u534f\u65b9\u5dee\u4e3a A=LLT\\\\mathbf{A} = \\\\mathbf{LL}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>LL<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u591a\u5143\u6b63\u6001\u5206\u5e03\u3002<\/p>\n

\u5728\u4f18\u5316\u7b97\u6cd5\u4e2d\u7684\u5e94\u7528<\/h4>\n

\u5728\u6570\u503c\u4f18\u5316\u4e2d，\u8bb8\u591a\u7b97\u6cd5\u9700\u8981\u6c42\u89e3\u5f62\u5982 Hx=\u2212g\\\\mathbf{Hx} = -\\\\mathbf{g}<\/span><\/span>Hx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>g<\/span><\/span><\/span><\/span><\/span> \u7684\u65b9\u7a0b，\u5176\u4e2d H\\\\mathbf{H}<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u662f Hessian \u77e9\u9635\u3002\u5bf9\u4e8e\u51f8\u4f18\u5316\u95ee\u9898，Hessian \u77e9\u9635\u662f\u5bf9\u79f0\u6b63\u5b9a\u7684，\u4f7f\u7528 Cholesky \u5206\u89e3\u6700\u4e3a\u9ad8\u6548\u3002\u725b\u987f\u6cd5\u7684\u6bcf\u6b21\u8fed\u4ee3\u9700\u8981\u6c42\u89e3\u8fd9\u6837\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u901a\u8fc7 Cholesky \u5206\u89e3，\u4e0d\u4ec5\u8ba1\u7b97\u6548\u7387\u9ad8，\u8fd8\u80fd\u68c0\u6d4b Hessian \u662f\u5426\u6b63\u5b9a\u2014\u2014\u5982\u679c\u5206\u89e3\u8fc7\u7a0b\u4e2d\u51fa\u73b0\u8d1f\u7684\u5bf9\u89d2\u5143\u7d20，\u8bf4\u660e\u5f53\u524d\u70b9\u4e0d\u662f\u4e25\u683c\u51f8\u7684，\u9700\u8981\u4fee\u6b63\u7b97\u6cd5\u3002<\/p>\n

\u5176\u4ed6\u91cd\u8981\u7684\u77e9\u9635\u5206\u89e3<\/h3>\n

Schur \u5206\u89e3<\/h4>\n

Schur \u5206\u89e3\u662f\u7279\u5f81\u503c\u5206\u89e3\u7684\u63a8\u5e7f，\u5bf9\u4efb\u610f\u65b9\u9635\u90fd\u5b58\u5728：<\/p>\n

A=QTQH\\\\mathbf{A} = \\\\mathbf{QTQ}^H<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>QTQ<\/span><\/span><\/span>H<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d Q\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> \u662f\u9149\u77e9\u9635，T\\\\mathbf{T}<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span> \u662f\u4e0a\u4e09\u89d2\u77e9\u9635，\u5bf9\u89d2\u7ebf\u5143\u7d20\u662f\u7279\u5f81\u503c\u3002\u4e0e Jordan \u6807\u51c6\u5f62\u76f8\u6bd4，Schur \u5206\u89e3\u5728\u6570\u503c\u4e0a\u66f4\u7a33\u5b9a，\u662f\u8bb8\u591a\u7279\u5f81\u503c\u7b97\u6cd5\u7684\u57fa\u7840\u3002<\/p>\n

\u5b9e Schur \u5206\u89e3\u7279\u522b\u91cd\u8981，\u5b83\u4fdd\u8bc1\u4e86\u5b9e\u77e9\u9635\u7684\u5206\u89e3\u7ed3\u679c\u4e5f\u662f\u5b9e\u7684，\u5176\u4e2d T\\\\mathbf{T}<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span> \u662f\u51c6\u4e0a\u4e09\u89d2\u77e9\u9635（\u5bf9\u89d2\u7ebf\u4e0a\u53ef\u80fd\u6709 2\u00d722 \\\\times 2<\/span><\/span>2<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span> \u7684\u5757\u5bf9\u5e94\u590d\u5171\u8f6d\u7279\u5f81\u503c\u5bf9）\u3002<\/p>\n

\u6781\u5206\u89e3<\/h4>\n

\u6781\u5206\u89e3\u5c06\u77e9\u9635\u5206\u89e3\u4e3a\u9149\u77e9\u9635\u548c\u534a\u6b63\u5b9a\u77e9\u9635\u7684\u4e58\u79ef：<\/p>\n

A=UP\\\\mathbf{A} = \\\\mathbf{UP}<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>UP<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span> \u662f\u9149\u77e9\u9635（\u4ee3\u8868\u65cb\u8f6c），P\\\\mathbf{P}<\/span><\/span>P<\/span><\/span><\/span><\/span><\/span> \u662f\u534a\u6b63\u5b9a Hermitian \u77e9\u9635（\u4ee3\u8868\u62c9\u4f38）\u3002\u8fd9\u79cd\u5206\u89e3\u5728\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\u7528\u4e8e\u5206\u79bb\u521a\u4f53\u53d8\u6362\u548c\u5f62\u53d8，\u5728\u529b\u5b66\u4e2d\u7528\u4e8e\u5206\u6790\u5e94\u53d8\u5f20\u91cf\u3002<\/p>\n

\u6781\u5206\u89e3\u4e0e SVD \u5bc6\u5207\u76f8\u5173：\u5982\u679c A=U\u03a3VT\\\\mathbf{A} = \\\\mathbf{U\\\\Sigma V}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3V<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f SVD，\u90a3\u4e48\u6781\u5206\u89e3\u4e3a A=(UVT)(V\u03a3VT)\\\\mathbf{A} = (\\\\mathbf{UV}^T)(\\\\mathbf{V\\\\Sigma V}^T)<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>UV<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>(<\/span>V\u03a3V<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u77e9\u9635\u5206\u89e3\u65b9\u6cd5\u7684\u9009\u62e9\u7b56\u7565<\/h3>\n

\u9009\u62e9\u5408\u9002\u7684\u77e9\u9635\u5206\u89e3\u65b9\u6cd5\u9700\u8981\u8003\u8651\u591a\u4e2a\u56e0\u7d20\u3002\u9996\u5148\u662f\u95ee\u9898\u7684\u6570\u5b66\u6027\u8d28：\u77e9\u9635\u662f\u5426\u5bf9\u79f0？\u662f\u5426\u6b63\u5b9a？\u662f\u5426\u7a00\u758f？\u6761\u4ef6\u6570\u5982\u4f55？\u5176\u6b21\u662f\u8ba1\u7b97\u9700\u6c42：\u9700\u8981\u5168\u90e8\u7279\u5f81\u503c\u8fd8\u662f\u90e8\u5206？\u662f\u4e00\u6b21\u6027\u6c42\u89e3\u8fd8\u662f\u591a\u6b21\u6c42\u89e3？\u5bf9\u7cbe\u5ea6\u548c\u901f\u5ea6\u7684\u8981\u6c42\u5982\u4f55？<\/p>\n

\u5bf9\u4e8e\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax=b\\\\mathbf{Ax} = \\\\mathbf{b}<\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>，\u5982\u679c A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u662f\u5bf9\u79f0\u6b63\u5b9a\u7684，Cholesky \u5206\u89e3\u662f\u6700\u4f73\u9009\u62e9，\u901f\u5ea6\u6700\u5feb\u4e14\u6570\u503c\u7a33\u5b9a\u3002\u5bf9\u4e8e\u4e00\u822c\u7684\u65b9\u9635，LU \u5206\u89e3\u914d\u5408\u90e8\u5206\u9009\u4e3b\u5143\u662f\u6807\u51c6\u65b9\u6cd5\u3002\u5982\u679c\u77e9\u9635\u63a5\u8fd1\u5947\u5f02\u6216\u6761\u4ef6\u6570\u5f88\u5927，\u5e94\u8be5\u4f7f\u7528 QR \u5206\u89e3\u6216 SVD \u4ee5\u83b7\u5f97\u66f4\u597d\u7684\u6570\u503c\u7a33\u5b9a\u6027\u3002<\/p>\n

\u5bf9\u4e8e\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898 min\u2061\u2225Ax\u2212b\u22252\\\\min \\\\|\\\\mathbf{Ax} – \\\\mathbf{b}\\\\|_2<\/span><\/span>min<\/span><\/span>\u2225<\/span>Ax<\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>b<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，QR \u5206\u89e3\u662f\u9996\u9009\u65b9\u6cd5，\u5b83\u5728\u4fdd\u6301\u6570\u503c\u7a33\u5b9a\u6027\u7684\u540c\u65f6\u6709\u8f83\u597d\u7684\u8ba1\u7b97\u6548\u7387\u3002\u5f53\u77e9\u9635\u4e25\u91cd\u75c5\u6001\u6216\u9700\u8981\u6b63\u5219\u5316\u65f6，SVD \u63d0\u4f9b\u4e86\u6700\u7a33\u5b9a\u7684\u89e3\u51b3\u65b9\u6848，\u53ef\u4ee5\u65b9\u4fbf\u5730\u5b9e\u73b0 Tikhonov \u6b63\u5219\u5316\u3002<\/p>\n

\u5728\u7279\u5f81\u503c\u95ee\u9898\u4e2d，\u5bf9\u4e8e\u4e2d\u5c0f\u89c4\u6a21\u7684\u5bc6\u96c6\u77e9\u9635，QR \u7b97\u6cd5\u8ba1\u7b97\u5168\u90e8\u7279\u5f81\u503c；\u5bf9\u4e8e\u5927\u89c4\u6a21\u7a00\u758f\u77e9\u9635，Lanczos \u6216 Arnoldi \u65b9\u6cd5\u8ba1\u7b97\u90e8\u5206\u7279\u5f81\u503c\u66f4\u9ad8\u6548\u3002\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u8ba1\u7b97\u901a\u5e38\u6bd4\u4e00\u822c\u77e9\u9635\u66f4\u5feb\u66f4\u7a33\u5b9a\u3002<\/p>\n

\u6570\u503c\u7a33\u5b9a\u6027\u7684\u6df1\u5165\u7406\u89e3<\/h3>\n

\u6570\u503c\u7a33\u5b9a\u6027\u662f\u9009\u62e9\u548c\u5b9e\u73b0\u77e9\u9635\u5206\u89e3\u7b97\u6cd5\u7684\u5173\u952e\u8003\u8651\u56e0\u7d20\u3002\u6761\u4ef6\u6570 \u03ba(A)=\u2225A\u2225\u2225A\u22121\u2225\\\\kappa(\\\\mathbf{A}) = \\\\|\\\\mathbf{A}\\\\| \\\\|\\\\mathbf{A}^{-1}\\\\|<\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225\u2225<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span><\/span><\/span><\/span><\/span> \u8861\u91cf\u4e86\u77e9\u9635\u5bf9\u6270\u52a8\u7684\u654f\u611f\u7a0b\u5ea6\u3002\u5f53 \u03ba(A)\u226b1\\\\kappa(\\\\mathbf{A}) \\\\gg 1<\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span>\u226b<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u65f6，\u77e9\u9635\u662f\u75c5\u6001\u7684，\u5c0f\u7684\u8f93\u5165\u8bef\u5dee\u4f1a\u5bfc\u81f4\u5927\u7684\u8f93\u51fa\u8bef\u5dee\u3002<\/p>\n

SVD \u76f4\u63a5\u63d0\u4f9b\u4e86\u6761\u4ef6\u6570\u7684\u8ba1\u7b97：\u03ba(A)=\u03c3max\u2061\/\u03c3min\u2061\\\\kappa(\\\\mathbf{A}) = \\\\sigma_{\\\\max}\/\\\\sigma_{\\\\min}<\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>m<\/span>a<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>\u03c3<\/span><\/span>m<\/span>i<\/span>n<\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48 SVD \u5728\u5904\u7406\u75c5\u6001\u95ee\u9898\u65f6\u7279\u522b\u6709\u7528\u2014\u2014\u5b83\u80fd\u591f\u8bc6\u522b\u5e76\u5904\u7406\u63a5\u8fd1\u96f6\u7684\u5947\u5f02\u503c，\u907f\u514d\u6570\u503c\u95ee\u9898\u3002<\/p>\n

\u5411\u540e\u7a33\u5b9a\u6027\u662f\u8bc4\u4ef7\u7b97\u6cd5\u7684\u91cd\u8981\u6807\u51c6\u3002\u4e00\u4e2a\u5411\u540e\u7a33\u5b9a\u7684\u7b97\u6cd5\u4ea7\u751f\u7684\u7ed3\u679c\u7b49\u4ef7\u4e8e\u5bf9\u7565\u5fae\u6270\u52a8\u7684\u8f93\u5165\u8fdb\u884c\u7cbe\u786e\u8ba1\u7b97\u3002Householder QR \u5206\u89e3\u662f\u5411\u540e\u7a33\u5b9a\u7684\u5178\u8303，\u800c\u7ecf\u5178 Gram-Schmidt \u5219\u4e0d\u662f\u3002\u8fd9\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u5b9e\u9645\u5e94\u7528\u4e2d\u4f18\u5148\u9009\u62e9 Householder \u65b9\u6cd5\u3002<\/p>\n

\u5728\u5b9e\u73b0\u4e2d，\u9009\u4e3b\u5143\u7b56\u7565\u5bf9\u63d0\u9ad8\u7a33\u5b9a\u6027\u81f3\u5173\u91cd\u8981\u3002\u90e8\u5206\u9009\u4e3b\u5143\u786e\u4fdd LU \u5206\u89e3\u4e2d\u7684\u4e58\u6570\u4e0d\u4f1a\u8fc7\u5927，\u5b8c\u5168\u9009\u4e3b\u5143\u867d\u7136\u7406\u8bba\u4e0a\u66f4\u7a33\u5b9a\u4f46\u8ba1\u7b97\u6210\u672c\u8f83\u9ad8\u3002\u5bf9\u4e8e\u5bf9\u79f0\u77e9\u9635，\u5bf9\u89d2\u9009\u4e3b\u5143\u53ef\u4ee5\u4fdd\u6301\u5bf9\u79f0\u6027\u540c\u65f6\u63d0\u9ad8\u7a33\u5b9a\u6027\u3002<\/p>\n

\u9644\u5f55：\u6838\u5fc3\u5b9a\u7406\u7684\u8be6\u7ec6\u63a8\u5bfc<\/h3>\n

A. \u5947\u5f02\u503c\u5206\u89e3\u7684\u5b58\u5728\u6027\u4e0e\u6784\u9020<\/h4>\n

\u5b9a\u7406 A.1（SVD \u5b58\u5728\u6027\u5b9a\u7406）：\u5bf9\u4e8e\u4efb\u610f m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u7684\u590d\u77e9\u9635 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span>，\u5b58\u5728 m\u00d7mm \\\\times m<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span> \u9149\u77e9\u9635 U\\\\mathbf{U}<\/span><\/span>U<\/span><\/span><\/span><\/span><\/span>，n\u00d7nn \\\\times n<\/span><\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u9149\u77e9\u9635 V\\\\mathbf{V}<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span>，\u4ee5\u53ca m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u5bf9\u89d2\u77e9\u9635 \u03a3\\\\mathbf{\\\\Sigma}<\/span><\/span>\u03a3<\/span><\/span><\/span><\/span><\/span>，\u4f7f\u5f97：
\nA=U\u03a3V\u2217\\\\mathbf{A} = \\\\mathbf{U\\\\Sigma V}^*<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3V<\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bc1\u660e：<\/p>\n

\u9996\u5148\u8003\u8651 n\u00d7nn \\\\times n<\/span><\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> Hermitian \u534a\u6b63\u5b9a\u77e9\u9635 A\u2217A\\\\mathbf{A}^*\\\\mathbf{A}<\/span><\/span>A<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span>\u3002\u7531\u8c31\u5b9a\u7406，\u5b58\u5728\u9149\u77e9\u9635 V\\\\mathbf{V}<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span> \u548c\u975e\u8d1f\u5b9e\u5bf9\u89d2\u77e9\u9635 D\\\\mathbf{D}<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span>，\u4f7f\u5f97：
\nA\u2217A=VDV\u2217\\\\mathbf{A}^*\\\\mathbf{A} = \\\\mathbf{VDV}^*<\/span><\/span>A<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>VDV<\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8bbe D=diag(\u03bb1,\u03bb2,\u2026,\u03bbn)\\\\mathbf{D} = \\\\text{diag}(\\\\lambda_1, \\\\lambda_2, \\\\ldots, \\\\lambda_n)<\/span><\/span>D<\/span><\/span>=<\/span><\/span><\/span><\/span>diag<\/span><\/span>(<\/span>\u03bb<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u03bb<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>\u03bb<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d \u03bb1\u2265\u03bb2\u2265\u2026\u2265\u03bbr>0=\u03bbr+1=\u2026=\u03bbn\\\\lambda_1 \\\\geq \\\\lambda_2 \\\\geq \\\\ldots \\\\geq \\\\lambda_r > 0 = \\\\lambda_{r+1} = \\\\ldots = \\\\lambda_n<\/span><\/span>\u03bb<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u2026<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>r<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2026<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，r=rank(A)r = \\\\text{rank}(\\\\mathbf{A})<\/span><\/span>r<\/span><\/span>=<\/span><\/span><\/span><\/span>rank<\/span><\/span>(<\/span>A<\/span>)<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u5b9a\u4e49\u5947\u5f02\u503c \u03c3i=\u03bbi\\\\sigma_i = \\\\sqrt{\\\\lambda_i}<\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5bf9\u4e8e i=1,\u2026,ri = 1, \\\\ldots, r<\/span><\/span>i<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>\u3002\u6784\u9020 m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635：
\n\u03a3=[diag(\u03c31,\u2026,\u03c3r)000]\\\\mathbf{\\\\Sigma} = \\\\begin{bmatrix} \\\\text{diag}(\\\\sigma_1, \\\\ldots, \\\\sigma_r) & \\\\mathbf{0} \\\\\\\\ \\\\mathbf{0} & \\\\mathbf{0} \\\\end{bmatrix}<\/span><\/span>\u03a3<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>diag<\/span><\/span>(<\/span>\u03c3<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>\u03c3<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee4 v1,\u2026,vn\\\\mathbf{v}_1, \\\\ldots, \\\\mathbf{v}_n<\/span><\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>v<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e3a V\\\\mathbf{V}<\/span><\/span>V<\/span><\/span><\/span><\/span><\/span> \u7684\u5217\u5411\u91cf（A\u2217A\\\\mathbf{A}^*\\\\mathbf{A}<\/span><\/span>A<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u7684\u7279\u5f81\u5411\u91cf）\u3002\u5bf9\u4e8e i=1,\u2026,ri = 1, \\\\ldots, r<\/span><\/span>i<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>，\u5b9a\u4e49：
\nui=1\u03c3iAvi\\\\mathbf{u}_i = \\\\frac{1}{\\\\sigma_i}\\\\mathbf{Av}_i<\/span><\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Av<\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u9700\u8981\u9a8c\u8bc1 {u1,\u2026,ur}\\\\{\\\\mathbf{u}_1, \\\\ldots, \\\\mathbf{u}_r\\\\}<\/span><\/span>{<\/span>u<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>u<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u662f\u6807\u51c6\u6b63\u4ea4\u7684\u3002\u5bf9\u4e8e i,j\u2264ri, j \\\\leq r<\/span><\/span>i<\/span>,<\/span><\/span>j<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>：
\nui\u2217uj=1\u03c3i\u03c3jvi\u2217A\u2217Avj=1\u03c3i\u03c3jvi\u2217(\u03bbjvj)=\u03bbj\u03c3i\u03c3jvi\u2217vj=\u03c3j2\u03c3i\u03c3j\u03b4ij=\u03b4ij\\\\mathbf{u}_i^*\\\\mathbf{u}_j = \\\\frac{1}{\\\\sigma_i\\\\sigma_j}\\\\mathbf{v}_i^*\\\\mathbf{A}^*\\\\mathbf{Av}_j = \\\\frac{1}{\\\\sigma_i\\\\sigma_j}\\\\mathbf{v}_i^*(\\\\lambda_j\\\\mathbf{v}_j) = \\\\frac{\\\\lambda_j}{\\\\sigma_i\\\\sigma_j}\\\\mathbf{v}_i^*\\\\mathbf{v}_j = \\\\frac{\\\\sigma_j^2}{\\\\sigma_i\\\\sigma_j}\\\\delta_{ij} = \\\\delta_{ij}<\/span><\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Av<\/span><\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>\u03bb<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>j<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span><\/span>ij<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u82e5 r<mr < m<\/span><\/span>r<\/span><\/span><<\/span><\/span><\/span><\/span>m<\/span><\/span><\/span><\/span><\/span>，\u901a\u8fc7 Gram-Schmidt \u8fc7\u7a0b\u6269\u5c55 {u1,\u2026,ur}\\\\{\\\\mathbf{u}_1, \\\\ldots, \\\\mathbf{u}_r\\\\}<\/span><\/span>{<\/span>u<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>u<\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u5230 Cm\\\\mathbb{C}^m<\/span><\/span>C<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u5b8c\u6574\u6807\u51c6\u6b63\u4ea4\u57fa {u1,\u2026,um}\\\\{\\\\mathbf{u}_1, \\\\ldots, \\\\mathbf{u}_m\\\\}<\/span><\/span>{<\/span>u<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>u<\/span><\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u6784\u9020 U=[u1\u2223\u2026\u2223um]\\\\mathbf{U} = [\\\\mathbf{u}_1 | \\\\ldots | \\\\mathbf{u}_m]<\/span><\/span>U<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>u<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span><\/span>\u2026<\/span><\/span>\u2223<\/span>u<\/span><\/span>m<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span>\u3002\u73b0\u5728\u9a8c\u8bc1 A=U\u03a3V\u2217\\\\mathbf{A} = \\\\mathbf{U\\\\Sigma V}^*<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3V<\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：<\/p>\n

\u5bf9\u4e8e i\u2264ri \\\\leq r<\/span><\/span>i<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>：
\nAvi=\u03c3iui=(U\u03a3)i\\\\mathbf{Av}_i = \\\\sigma_i\\\\mathbf{u}_i = (\\\\mathbf{U\\\\Sigma})_i<\/span><\/span>Av<\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>U\u03a3<\/span><\/span>)<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5bf9\u4e8e i>ri > r<\/span><\/span>i<\/span><\/span>><\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span>：
\nAvi=0=(U\u03a3)i\\\\mathbf{Av}_i = \\\\mathbf{0} = (\\\\mathbf{U\\\\Sigma})_i<\/span><\/span>Av<\/span><\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>U\u03a3<\/span><\/span>)<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64 AV=U\u03a3\\\\mathbf{AV} = \\\\mathbf{U\\\\Sigma}<\/span><\/span>AV<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3<\/span><\/span><\/span><\/span><\/span><\/span>，\u5373 A=U\u03a3V\u2217\\\\mathbf{A} = \\\\mathbf{U\\\\Sigma V}^*<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3V<\/span><\/span><\/span>\u2217<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

B. Eckart-Young-Mirsky \u5b9a\u7406：\u6700\u4f18\u4f4e\u79e9\u8fd1\u4f3c<\/h4>\n

\u5b9a\u7406 B.1：\u8bbe A=U\u03a3VT\\\\mathbf{A} = \\\\mathbf{U\\\\Sigma V}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>U\u03a3V<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635\u7684 SVD，\u5176\u4e2d\u5947\u5f02\u503c\u4e3a \u03c31\u2265\u03c32\u2265\u2026\u2265\u03c3min\u2061(m,n)\\\\sigma_1 \\\\geq \\\\sigma_2 \\\\geq \\\\ldots \\\\geq \\\\sigma_{\\\\min(m,n)}<\/span><\/span>\u03c3<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u2026<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>m<\/span>i<\/span>n<\/span><\/span>(<\/span>m<\/span>,<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u5bf9\u4e8e k<rank(A)k < \\\\text{rank}(\\\\mathbf{A})<\/span><\/span>k<\/span><\/span><<\/span><\/span><\/span><\/span>rank<\/span><\/span>(<\/span>A<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u5b9a\u4e49：
\nAk=\u2211i=1k\u03c3iuiviT\\\\mathbf{A}_k = \\\\sum_{i=1}^k \\\\sigma_i\\\\mathbf{u}_i\\\\mathbf{v}_i^T<\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5219\u5bf9\u4e8e\u4efb\u610f\u79e9\u4e0d\u8d85\u8fc7 kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u7684\u77e9\u9635 B\\\\mathbf{B}<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span>：
\n\u2225A\u2212Ak\u2225F\u2264\u2225A\u2212B\u2225F\\\\|\\\\mathbf{A} – \\\\mathbf{A}_k\\\\|_F \\\\leq \\\\|\\\\mathbf{A} – \\\\mathbf{B}\\\\|_F<\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span><\/span>F<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>B<\/span>\u2225<\/span><\/span>F<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d \u2225\u22c5\u2225F\\\\|\\\\cdot\\\\|_F<\/span><\/span>\u2225<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>\u2225<\/span><\/span>F<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f Frobenius \u8303\u6570\u3002<\/p>\n

\u8bc1\u660e：<\/p>\n

\u9996\u5148\u6ce8\u610f\u5230：
\n\u2225A\u2212Ak\u2225F2=\u2225\u2211i=k+1r\u03c3iuiviT\u2225F2=\u2211i=k+1r\u03c3i2\\\\|\\\\mathbf{A} – \\\\mathbf{A}_k\\\\|_F^2 = \\\\left\\\\|\\\\sum_{i=k+1}^r \\\\sigma_i\\\\mathbf{u}_i\\\\mathbf{v}_i^T\\\\right\\\\|_F^2 = \\\\sum_{i=k+1}^r \\\\sigma_i^2<\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span><\/span>F<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>F<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>r<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u662f\u56e0\u4e3a {uiviT}\\\\{\\\\mathbf{u}_i\\\\mathbf{v}_i^T\\\\}<\/span><\/span>{<\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u6784\u6210\u6b63\u4ea4\u57fa\u3002<\/p>\n

\u8bbe B\\\\mathbf{B}<\/span><\/span>B<\/span><\/span><\/span><\/span><\/span> \u662f\u4efb\u610f\u79e9\u4e0d\u8d85\u8fc7 kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u7684\u77e9\u9635\u3002\u5b58\u5728\u7ef4\u6570\u81f3\u5c11\u4e3a n\u2212kn-k<\/span><\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u7684\u5b50\u7a7a\u95f4 V\u2286Rn\\\\mathcal{V} \\\\subseteq \\\\mathbb{R}^n<\/span><\/span>V<\/span><\/span>\u2286<\/span><\/span><\/span><\/span>R<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4f7f\u5f97 Bx=0\\\\mathbf{Bx} = \\\\mathbf{0}<\/span><\/span>Bx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u5bf9\u6240\u6709 x\u2208V\\\\mathbf{x} \\\\in \\\\mathcal{V}<\/span><\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u8003\u8651\u7531 {v1,\u2026,vk+1}\\\\{\\\\mathbf{v}_1, \\\\ldots, \\\\mathbf{v}_{k+1}\\\\}<\/span><\/span>{<\/span>v<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>v<\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>}<\/span><\/span><\/span><\/span><\/span> \u5f20\u6210\u7684 (k+1)(k+1)<\/span><\/span>(<\/span>k<\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span> \u7ef4\u5b50\u7a7a\u95f4 W\\\\mathcal{W}<\/span><\/span>W<\/span><\/span><\/span><\/span><\/span>\u3002\u7531\u4e8e dim\u2061(V)+dim\u2061(W)=(n\u2212k)+(k+1)=n+1>n\\\\dim(\\\\mathcal{V}) + \\\\dim(\\\\mathcal{W}) = (n-k) + (k+1) = n+1 > n<\/span><\/span>dim<\/span>(<\/span>V<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>dim<\/span>(<\/span>W<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>k<\/span>)<\/span><\/span>+<\/span><\/span><\/span><\/span>(<\/span>k<\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>n<\/span><\/span>+<\/span><\/span><\/span><\/span>1<\/span><\/span>><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>，\u5fc5\u5b58\u5728\u975e\u96f6\u5411\u91cf x\u2208V\u2229W\\\\mathbf{x} \\\\in \\\\mathcal{V} \\\\cap \\\\mathcal{W}<\/span><\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>V<\/span><\/span>\u2229<\/span><\/span><\/span><\/span>W<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u4e0d\u5931\u4e00\u822c\u6027，\u8bbe \u2225x\u2225=1\\\\|\\\\mathbf{x}\\\\| = 1<\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002\u7531\u4e8e x\u2208W\\\\mathbf{x} \\\\in \\\\mathcal{W}<\/span><\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>W<\/span><\/span><\/span><\/span><\/span>，\u53ef\u4ee5\u5199\u6210：
\nx=\u2211i=1k+1\u03b1ivi,\u2211i=1k+1\u03b1i2=1\\\\mathbf{x} = \\\\sum_{i=1}^{k+1} \\\\alpha_i\\\\mathbf{v}_i, \\\\quad \\\\sum_{i=1}^{k+1} \\\\alpha_i^2 = 1<\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：
\nAx=\u2211i=1k+1\u03b1i\u03c3iui\\\\mathbf{Ax} = \\\\sum_{i=1}^{k+1} \\\\alpha_i\\\\sigma_i\\\\mathbf{u}_i<\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>u<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531\u4e8e x\u2208V\\\\mathbf{x} \\\\in \\\\mathcal{V}<\/span><\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span>，\u6709 Bx=0\\\\mathbf{Bx} = \\\\mathbf{0}<\/span><\/span>Bx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u6240\u4ee5：
\n\u2225(A\u2212B)x\u22252=\u2225Ax\u22252=\u2211i=1k+1\u03b1i2\u03c3i2\u2265\u03c3k+12\u2211i=1k+1\u03b1i2=\u03c3k+12\\\\|(\\\\mathbf{A} – \\\\mathbf{B})\\\\mathbf{x}\\\\|^2 = \\\\|\\\\mathbf{Ax}\\\\|^2 = \\\\sum_{i=1}^{k+1} \\\\alpha_i^2\\\\sigma_i^2 \\\\geq \\\\sigma_{k+1}^2\\\\sum_{i=1}^{k+1} \\\\alpha_i^2 = \\\\sigma_{k+1}^2<\/span><\/span>\u2225<\/span>(<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>B<\/span>)<\/span>x<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>Ax<\/span><\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>i<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531 Frobenius \u8303\u6570\u7684\u53d8\u5206\u7279\u5f81：
\n\u2225A\u2212B\u2225F2\u2265\u03c3k+12+\u03c3k+22+\u2026+\u03c3r2=\u2225A\u2212Ak\u2225F2\\\\|\\\\mathbf{A} – \\\\mathbf{B}\\\\|_F^2 \\\\geq \\\\sigma_{k+1}^2 + \\\\sigma_{k+2}^2 + \\\\ldots + \\\\sigma_r^2 = \\\\|\\\\mathbf{A} – \\\\mathbf{A}_k\\\\|_F^2<\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>B<\/span>\u2225<\/span><\/span>F<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>k<\/span>+<\/span>1<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>k<\/span>+<\/span>2<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u2026<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>r<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>A<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span><\/span>F<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u5b8c\u6210\u4e86\u8bc1\u660e\u3002<\/p>\n

C. QR \u5206\u89e3\u7684 Householder \u6784\u9020<\/h4>\n

\u5b9a\u7406 C.1：\u5bf9\u4e8e\u4efb\u610f m\u00d7nm \\\\times n<\/span><\/span>m<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span>（m\u2265nm \\\\geq n<\/span><\/span>m<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>），\u5b58\u5728\u6b63\u4ea4\u77e9\u9635 Q\\\\mathbf{Q}<\/span><\/span>Q<\/span><\/span><\/span><\/span><\/span> \u548c\u4e0a\u4e09\u89d2\u77e9\u9635 R\\\\mathbf{R}<\/span><\/span>R<\/span><\/span><\/span><\/span><\/span> \u4f7f\u5f97 A=QR\\\\mathbf{A} = \\\\mathbf{QR}<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>QR<\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u6784\u9020\u6027\u8bc1\u660e：<\/p>\n

Householder \u53cd\u5c04\u5668\u5b9a\u4e49\u4e3a：
\nH=I\u22122vvTvTv\\\\mathbf{H} = \\\\mathbf{I} – 2\\\\frac{\\\\mathbf{vv}^T}{\\\\mathbf{v}^T\\\\mathbf{v}}<\/span><\/span>H<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>vv<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d v\u22600\\\\mathbf{v} \\\\neq \\\\mathbf{0}<\/span><\/span>v<\/span><\/span><\/span>\ue020<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span> \u662f\u53cd\u5c04\u5411\u91cf\u3002H\\\\mathbf{H}<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u662f\u5bf9\u79f0\u6b63\u4ea4\u77e9\u9635，\u6ee1\u8db3 H2=I\\\\mathbf{H}^2 = \\\\mathbf{I}<\/span><\/span>H<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u7ed9\u5b9a\u5411\u91cf x\u2208Rm\\\\mathbf{x} \\\\in \\\\mathbb{R}^m<\/span><\/span>x<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>R<\/span><\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8981\u627e\u5230 H\\\\mathbf{H}<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u4f7f\u5f97 Hx=\u03b1e1\\\\mathbf{Hx} = \\\\alpha\\\\mathbf{e}_1<\/span><\/span>Hx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span>e<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d e1\\\\mathbf{e}_1<\/span><\/span>e<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u7b2c\u4e00\u4e2a\u6807\u51c6\u57fa\u5411\u91cf，\u03b1=\u00b1\u2225x\u2225\\\\alpha = \\\\pm\\\\|\\\\mathbf{x}\\\\|<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span>\u00b1<\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u8bbe v=x\u2212\u03b1e1\\\\mathbf{v} = \\\\mathbf{x} – \\\\alpha\\\\mathbf{e}_1<\/span><\/span>v<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span>e<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u5219：
\nHx=x\u22122vTxvTvv\\\\mathbf{Hx} = \\\\mathbf{x} – 2\\\\frac{\\\\mathbf{v}^T\\\\mathbf{x}}{\\\\mathbf{v}^T\\\\mathbf{v}}\\\\mathbf{v}<\/span><\/span>Hx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8ba1\u7b97：
\nvTx=(x\u2212\u03b1e1)Tx=\u2225x\u22252\u2212\u03b1x1\\\\mathbf{v}^T\\\\mathbf{x} = (\\\\mathbf{x} – \\\\alpha\\\\mathbf{e}_1)^T\\\\mathbf{x} = \\\\|\\\\mathbf{x}\\\\|^2 – \\\\alpha x_1<\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span>e<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\nvTv=\u2225x\u22252\u22122\u03b1x1+\u03b12=2(\u03b12\u2212\u03b1x1)\\\\mathbf{v}^T\\\\mathbf{v} = \\\\|\\\\mathbf{x}\\\\|^2 – 2\\\\alpha x_1 + \\\\alpha^2 = 2(\\\\alpha^2 – \\\\alpha x_1)<\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span>\u03b1<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>2<\/span>(<\/span>\u03b1<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03b1<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：
\nvTxvTv=\u03b12\u2212\u03b1x12(\u03b12\u2212\u03b1x1)=12\\\\frac{\\\\mathbf{v}^T\\\\mathbf{x}}{\\\\mathbf{v}^T\\\\mathbf{v}} = \\\\frac{\\\\alpha^2 – \\\\alpha x_1}{2(\\\\alpha^2 – \\\\alpha x_1)} = \\\\frac{1}{2}<\/span><\/span><\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span><\/span><\/span><\/span><\/span><\/span>v<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span>(<\/span>\u03b1<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u03b1<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>\u03b1<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5：
\nHx=x\u2212v=\u03b1e1\\\\mathbf{Hx} = \\\\mathbf{x} – \\\\mathbf{v} = \\\\alpha\\\\mathbf{e}_1<\/span><\/span>Hx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>v<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b1<\/span>e<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e3a\u4e86\u6570\u503c\u7a33\u5b9a\u6027，\u9009\u62e9 \u03b1=\u2212sign(x1)\u2225x\u2225\\\\alpha = -\\\\text{sign}(x_1)\\\\|\\\\mathbf{x}\\\\|<\/span><\/span>\u03b1<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2212<\/span>sign<\/span><\/span>(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span> \u4ee5\u907f\u514d\u6d88\u53bb\u3002<\/p>\n

QR \u5206\u89e3\u7b97\u6cd5：<\/p>\n

\u5bf9\u77e9\u9635 A(0)=A\\\\mathbf{A}^{(0)} = \\\\mathbf{A}<\/span><\/span>A<\/span><\/span>(<\/span>0<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u8fdb\u884c\u8fed\u4ee3\u3002\u5728\u7b2c kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u6b65，\u8bbe：
\nA(k\u22121)=[R11(k\u22121)R12(k\u22121)0A22(k\u22121)]\\\\mathbf{A}^{(k-1)} = \\\\begin{bmatrix} \\\\mathbf{R}_{11}^{(k-1)} & \\\\mathbf{R}_{12}^{(k-1)} \\\\\\\\ \\\\mathbf{0} & \\\\mathbf{A}_{22}^{(k-1)} \\\\end{bmatrix}<\/span><\/span>A<\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>R<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>12<\/span><\/span><\/span><\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d R11(k\u22121)\\\\mathbf{R}_{11}^{(k-1)}<\/span><\/span>R<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f (k\u22121)\u00d7(k\u22121)(k-1) \\\\times (k-1)<\/span><\/span>(<\/span>k<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>(<\/span>k<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span> \u4e0a\u4e09\u89d2\u77e9\u9635\u3002<\/p>\n

\u6784\u9020 Householder \u77e9\u9635 Hk\\\\mathbf{H}_k<\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5c06 A22(k\u22121)\\\\mathbf{A}_{22}^{(k-1)}<\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u7b2c\u4e00\u5217\u5316\u4e3a \u03b1e1\\\\alpha\\\\mathbf{e}_1<\/span><\/span>\u03b1<\/span>e<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>：
\nQk=[Ik\u2212100Hk]\\\\mathbf{Q}_k = \\\\begin{bmatrix} \\\\mathbf{I}_{k-1} & \\\\mathbf{0} \\\\\\\\ \\\\mathbf{0} & \\\\mathbf{H}_k \\\\end{bmatrix}<\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>I<\/span><\/span>k<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ee4 A(k)=QkA(k\u22121)\\\\mathbf{A}^{(k)} = \\\\mathbf{Q}_k\\\\mathbf{A}^{(k-1)}<\/span><\/span>A<\/span><\/span>(<\/span>k<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>(<\/span>k<\/span>\u2212<\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u7ecf\u8fc7 nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u6b65\u540e，A(n)=R\\\\mathbf{A}^{(n)} = \\\\mathbf{R}<\/span><\/span>A<\/span><\/span>(<\/span>n<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>R<\/span><\/span><\/span><\/span><\/span> \u662f\u4e0a\u4e09\u89d2\u77e9\u9635，\u4e14：
\nR=QnQn\u22121\u22efQ1A\\\\mathbf{R} = \\\\mathbf{Q}_n\\\\mathbf{Q}_{n-1}\\\\cdots\\\\mathbf{Q}_1\\\\mathbf{A}<\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>Q<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：
\nA=Q1TQ2T\u22efQnTR=QR\\\\mathbf{A} = \\\\mathbf{Q}_1^T\\\\mathbf{Q}_2^T\\\\cdots\\\\mathbf{Q}_n^T\\\\mathbf{R} = \\\\mathbf{QR}<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>1<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>2<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>Q<\/span><\/span>n<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>R<\/span><\/span>=<\/span><\/span><\/span><\/span>QR<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d Q=Q1TQ2T\u22efQnT\\\\mathbf{Q} = \\\\mathbf{Q}_1^T\\\\mathbf{Q}_2^T\\\\cdots\\\\mathbf{Q}_n^T<\/span><\/span>Q<\/span><\/span>=<\/span><\/span><\/span><\/span>Q<\/span><\/span>1<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Q<\/span><\/span>2<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22ef<\/span><\/span>Q<\/span><\/span>n<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u4ea4\u77e9\u9635\u3002<\/p>\n

D. Cholesky \u5206\u89e3\u7684\u552f\u4e00\u6027\u4e0e\u7a33\u5b9a\u6027<\/h4>\n

\u5b9a\u7406 D.1：\u82e5 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u662f n\u00d7nn \\\\times n<\/span><\/span>n<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u5bf9\u79f0\u6b63\u5b9a\u77e9\u9635，\u5219\u5b58\u5728\u552f\u4e00\u7684\u4e0b\u4e09\u89d2\u77e9\u9635 L\\\\mathbf{L}<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span>（\u5bf9\u89d2\u5143\u7d20\u4e3a\u6b63）\u4f7f\u5f97 A=LLT\\\\mathbf{A} = \\\\mathbf{LL}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>LL<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u8bc1\u660e：<\/p>\n

\u5b58\u5728\u6027：\u4f7f\u7528\u5f52\u7eb3\u6cd5\u3002<\/p>\n

\u5bf9\u4e8e n=1n = 1<\/span><\/span>n<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>，A=[a]\\\\mathbf{A} = [a]<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span>a<\/span>]<\/span><\/span><\/span><\/span><\/span> \u5176\u4e2d a>0a > 0<\/span><\/span>a<\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u53d6 L=[a]\\\\mathbf{L} = [\\\\sqrt{a}]<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u5047\u8bbe\u5b9a\u7406\u5bf9 (n\u22121)\u00d7(n\u22121)(n-1) \\\\times (n-1)<\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635\u6210\u7acb\u3002\u5c06 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u5206\u5757\u4e3a：
\nA=[a11aTaA22]\\\\mathbf{A} = \\\\begin{bmatrix} a_{11} & \\\\mathbf{a}^T \\\\\\\\ \\\\mathbf{a} & \\\\mathbf{A}_{22} \\\\end{bmatrix}<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d a11>0a_{11} > 0<\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>（\u56e0\u4e3a A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u6b63\u5b9a），a\u2208Rn\u22121\\\\mathbf{a} \\\\in \\\\mathbb{R}^{n-1}<\/span><\/span>a<\/span><\/span>\u2208<\/span><\/span><\/span><\/span>R<\/span><\/span>n<\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，A22\\\\mathbf{A}_{22}<\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f (n\u22121)\u00d7(n\u22121)(n-1) \\\\times (n-1)<\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span>\u00d7<\/span><\/span><\/span><\/span>(<\/span>n<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>1<\/span>)<\/span><\/span><\/span><\/span><\/span> \u77e9\u9635\u3002<\/p>\n

\u8bbe：
\nL=[l110TlL22]\\\\mathbf{L} = \\\\begin{bmatrix} l_{11} & \\\\mathbf{0}^T \\\\\\\\ \\\\mathbf{l} & \\\\mathbf{L}_{22} \\\\end{bmatrix}<\/span><\/span>L<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>l<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>0<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece A=LLT\\\\mathbf{A} = \\\\mathbf{LL}^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>LL<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5f97：
\na11=l112\u21d2l11=a11a_{11} = l_{11}^2 \\\\Rightarrow l_{11} = \\\\sqrt{a_{11}}<\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>l<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u21d2<\/span><\/span><\/span><\/span>l<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
\na=l11l\u21d2l=1l11a\\\\mathbf{a} = l_{11}\\\\mathbf{l} \\\\Rightarrow \\\\mathbf{l} = \\\\frac{1}{l_{11}}\\\\mathbf{a}<\/span><\/span>a<\/span><\/span>=<\/span><\/span><\/span><\/span>l<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>\u21d2<\/span><\/span><\/span><\/span>l<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>l<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span><\/span><\/span><\/span><\/span>
\nA22=llT+L22L22T\\\\mathbf{A}_{22} = \\\\mathbf{ll}^T + \\\\mathbf{L}_{22}\\\\mathbf{L}_{22}^T<\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>ll<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：
\nL22L22T=A22\u2212llT=A22\u22121a11aaT\\\\mathbf{L}_{22}\\\\mathbf{L}_{22}^T = \\\\mathbf{A}_{22} – \\\\mathbf{ll}^T = \\\\mathbf{A}_{22} – \\\\frac{1}{a_{11}}\\\\mathbf{aa}^T<\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>ll<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>aa<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u9700\u8981\u8bc1\u660e S=A22\u22121a11aaT\\\\mathbf{S} = \\\\mathbf{A}_{22} – \\\\frac{1}{a_{11}}\\\\mathbf{aa}^T<\/span><\/span>S<\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>aa<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u5b9a\u7684\u3002\u5bf9\u4e8e\u4efb\u610f x\u22600\\\\mathbf{x} \\\\neq \\\\mathbf{0}<\/span><\/span>x<\/span><\/span><\/span>\ue020<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>：
\nxTSx=xTA22x\u22121a11(aTx)2\\\\mathbf{x}^T\\\\mathbf{Sx} = \\\\mathbf{x}^T\\\\mathbf{A}_{22}\\\\mathbf{x} – \\\\frac{1}{a_{11}}(\\\\mathbf{a}^T\\\\mathbf{x})^2<\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Sx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8003\u8651\u5411\u91cf y=[\u2212aTxa11x]\\\\mathbf{y} = \\\\begin{bmatrix} -\\\\frac{\\\\mathbf{a}^T\\\\mathbf{x}}{a_{11}} \\\\\\\\ \\\\mathbf{x} \\\\end{bmatrix}<\/span><\/span>y<\/span><\/span>=<\/span><\/span><\/span><\/span>[<\/span><\/span><\/span>\u2212<\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u5219：
\nyTAy=a11(\u2212aTxa11)2\u22122aTxa11aTx+xTA22x\\\\mathbf{y}^T\\\\mathbf{Ay} = a_{11}\\\\left(-\\\\frac{\\\\mathbf{a}^T\\\\mathbf{x}}{a_{11}}\\\\right)^2 – 2\\\\frac{\\\\mathbf{a}^T\\\\mathbf{x}}{a_{11}}\\\\mathbf{a}^T\\\\mathbf{x} + \\\\mathbf{x}^T\\\\mathbf{A}_{22}\\\\mathbf{x}<\/span><\/span>y<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Ay<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span>\u2212<\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span><\/span><\/span><\/span><\/span>
\n=(aTx)2a11\u22122(aTx)2a11+xTA22x=xTSx= \\\\frac{(\\\\mathbf{a}^T\\\\mathbf{x})^2}{a_{11}} – 2\\\\frac{(\\\\mathbf{a}^T\\\\mathbf{x})^2}{a_{11}} + \\\\mathbf{x}^T\\\\mathbf{A}_{22}\\\\mathbf{x} = \\\\mathbf{x}^T\\\\mathbf{Sx}<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span>2<\/span><\/span><\/span>a<\/span><\/span>11<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>a<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Sx<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7531\u4e8e A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u6b63\u5b9a\u4e14 y\u22600\\\\mathbf{y} \\\\neq \\\\mathbf{0}<\/span><\/span>y<\/span><\/span><\/span>\ue020<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>，\u6545 xTSx=yTAy>0\\\\mathbf{x}^T\\\\mathbf{Sx} = \\\\mathbf{y}^T\\\\mathbf{Ay} > 0<\/span><\/span>x<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Sx<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>y<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Ay<\/span><\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u56e0\u6b64 S\\\\mathbf{S}<\/span><\/span>S<\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u5b9a\u7684，\u7531\u5f52\u7eb3\u5047\u8bbe，\u5b58\u5728\u552f\u4e00\u7684 L22\\\\mathbf{L}_{22}<\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4f7f\u5f97 S=L22L22T\\\\mathbf{S} = \\\\mathbf{L}_{22}\\\\mathbf{L}_{22}^T<\/span><\/span>S<\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>22<\/span><\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

\u552f\u4e00\u6027：\u5047\u8bbe A=L1L1T=L2L2T\\\\mathbf{A} = \\\\mathbf{L}_1\\\\mathbf{L}_1^T = \\\\mathbf{L}_2\\\\mathbf{L}_2^T<\/span><\/span>A<\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d L1,L2\\\\mathbf{L}_1, \\\\mathbf{L}_2<\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u90fd\u662f\u5bf9\u89d2\u5143\u7d20\u4e3a\u6b63\u7684\u4e0b\u4e09\u89d2\u77e9\u9635\u3002<\/p>\n

\u4ee4 M=L1L2\u22121\\\\mathbf{M} = \\\\mathbf{L}_1\\\\mathbf{L}_2^{-1}<\/span><\/span>M<\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u5219：
\nMMT=L1L2\u22121(L2\u22121)TL1T=L1L1T(L2T)\u22121L2\u22121=AA\u22121=I\\\\mathbf{MM}^T = \\\\mathbf{L}_1\\\\mathbf{L}_2^{-1}(\\\\mathbf{L}_2^{-1})^T\\\\mathbf{L}_1^T = \\\\mathbf{L}_1\\\\mathbf{L}_1^T(\\\\mathbf{L}_2^T)^{-1}\\\\mathbf{L}_2^{-1} = \\\\mathbf{A}\\\\mathbf{A}^{-1} = \\\\mathbf{I}<\/span><\/span>MM<\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u6240\u4ee5 M\\\\mathbf{M}<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u4ea4\u77e9\u9635\u3002\u4f46 M\\\\mathbf{M}<\/span><\/span>M<\/span><\/span><\/span><\/span><\/span> \u540c\u65f6\u662f\u4e0b\u4e09\u89d2\u77e9\u9635（\u4e24\u4e2a\u4e0b\u4e09\u89d2\u77e9\u9635\u4e4b\u79ef），\u4e14\u5bf9\u89d2\u5143\u7d20\u4e3a\u6b63（mii=l1,ii\/l2,ii>0m_{ii} = l_{1,ii}\/l_{2,ii} > 0<\/span><\/span>m<\/span><\/span>ii<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>l<\/span><\/span>1<\/span>,<\/span>ii<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/<\/span>l<\/span><\/span>2<\/span>,<\/span>ii<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>><\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>）\u3002<\/p>\n

\u6b63\u4ea4\u4e0b\u4e09\u89d2\u77e9\u9635\u5fc5\u4e3a\u5bf9\u89d2\u77e9\u9635，\u4e14\u5bf9\u89d2\u5143\u7d20\u4e3a \u00b11\\\\pm 1<\/span><\/span>\u00b1<\/span>1<\/span><\/span><\/span><\/span><\/span>\u3002\u7531\u4e8e\u5bf9\u89d2\u5143\u7d20\u4e3a\u6b63，\u6545 M=I\\\\mathbf{M} = \\\\mathbf{I}<\/span><\/span>M<\/span><\/span>=<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span><\/span><\/span>，\u5373 L1=L2\\\\mathbf{L}_1 = \\\\mathbf{L}_2<\/span><\/span>L<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>L<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n

E. \u6761\u4ef6\u6570\u4e0e\u6270\u52a8\u5206\u6790<\/h4>\n

\u5b9a\u7406 E.1（\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6270\u52a8\u754c）：\u8bbe Ax=b\\\\mathbf{Ax} = \\\\mathbf{b}<\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>，A(x+\u03b4x)=b+\u03b4b\\\\mathbf{A}(\\\\mathbf{x} + \\\\delta\\\\mathbf{x}) = \\\\mathbf{b} + \\\\delta\\\\mathbf{b}<\/span><\/span>A<\/span>(<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span>b<\/span><\/span><\/span><\/span><\/span>\u3002\u5219：
\n\u2225\u03b4x\u2225\u2225x\u2225\u2264\u03ba(A)\u2225\u03b4b\u2225\u2225b\u2225\\\\frac{\\\\|\\\\delta\\\\mathbf{x}\\\\|}{\\\\|\\\\mathbf{x}\\\\|} \\\\leq \\\\kappa(\\\\mathbf{A})\\\\frac{\\\\|\\\\delta\\\\mathbf{b}\\\\|}{\\\\|\\\\mathbf{b}\\\\|}<\/span><\/span><\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span><\/span>\u2225<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d \u03ba(A)=\u2225A\u2225\u2225A\u22121\u2225\\\\kappa(\\\\mathbf{A}) = \\\\|\\\\mathbf{A}\\\\|\\\\|\\\\mathbf{A}^{-1}\\\\|<\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225\u2225<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span><\/span><\/span><\/span><\/span> \u662f\u6761\u4ef6\u6570\u3002<\/p>\n

\u8bc1\u660e：<\/p>\n

\u4ece A\u03b4x=\u03b4b\\\\mathbf{A}\\\\delta\\\\mathbf{x} = \\\\delta\\\\mathbf{b}<\/span><\/span>A<\/span>\u03b4<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>\u03b4<\/span>b<\/span><\/span><\/span><\/span><\/span> \u5f97：
\n\u03b4x=A\u22121\u03b4b\\\\delta\\\\mathbf{x} = \\\\mathbf{A}^{-1}\\\\delta\\\\mathbf{b}<\/span><\/span>\u03b4<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b4<\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u56e0\u6b64：
\n\u2225\u03b4x\u2225\u2264\u2225A\u22121\u2225\u2225\u03b4b\u2225\\\\|\\\\delta\\\\mathbf{x}\\\\| \\\\leq \\\\|\\\\mathbf{A}^{-1}\\\\|\\\\|\\\\delta\\\\mathbf{b}\\\\|<\/span><\/span>\u2225<\/span>\u03b4<\/span>x<\/span>\u2225<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225\u2225<\/span>\u03b4<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4ece Ax=b\\\\mathbf{Ax} = \\\\mathbf{b}<\/span><\/span>Ax<\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u5f97：
\n\u2225b\u2225\u2264\u2225A\u2225\u2225x\u2225\\\\|\\\\mathbf{b}\\\\| \\\\leq \\\\|\\\\mathbf{A}\\\\|\\\\|\\\\mathbf{x}\\\\|<\/span><\/span>\u2225<\/span>b<\/span>\u2225<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5373：
\n1\u2225x\u2225\u2264\u2225A\u2225\u2225b\u2225\\\\frac{1}{\\\\|\\\\mathbf{x}\\\\|} \\\\leq \\\\frac{\\\\|\\\\mathbf{A}\\\\|}{\\\\|\\\\mathbf{b}\\\\|}<\/span><\/span><\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u7ed3\u5408\u4e24\u4e2a\u4e0d\u7b49\u5f0f：
\n\u2225\u03b4x\u2225\u2225x\u2225\u2264\u2225A\u22121\u2225\u2225\u03b4b\u2225\u22c5\u2225A\u2225\u2225b\u2225=\u03ba(A)\u2225\u03b4b\u2225\u2225b\u2225\\\\frac{\\\\|\\\\delta\\\\mathbf{x}\\\\|}{\\\\|\\\\mathbf{x}\\\\|} \\\\leq \\\\|\\\\mathbf{A}^{-1}\\\\|\\\\|\\\\delta\\\\mathbf{b}\\\\| \\\\cdot \\\\frac{\\\\|\\\\mathbf{A}\\\\|}{\\\\|\\\\mathbf{b}\\\\|} = \\\\kappa(\\\\mathbf{A})\\\\frac{\\\\|\\\\delta\\\\mathbf{b}\\\\|}{\\\\|\\\\mathbf{b}\\\\|}<\/span><\/span><\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225\u2225<\/span>\u03b4<\/span>b<\/span>\u2225<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span><\/span>\u2225<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u4e2a\u754c\u662f\u7d27\u7684：\u5b58\u5728 b\\\\mathbf{b}<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u548c \u03b4b\\\\delta\\\\mathbf{b}<\/span><\/span>\u03b4<\/span>b<\/span><\/span><\/span><\/span><\/span> \u4f7f\u7b49\u53f7\u6210\u7acb\u3002<\/p>\n

\u63a8\u8bba E.1：\u82e5\u540c\u65f6\u8003\u8651 A\\\\mathbf{A}<\/span><\/span>A<\/span><\/span><\/span><\/span><\/span> \u548c b\\\\mathbf{b}<\/span><\/span>b<\/span><\/span><\/span><\/span><\/span> \u7684\u6270\u52a8：
\n(A+\u03b4A)(x+\u03b4x)=b+\u03b4b(\\\\mathbf{A} + \\\\delta\\\\mathbf{A})(\\\\mathbf{x} + \\\\delta\\\\mathbf{x}) = \\\\mathbf{b} + \\\\delta\\\\mathbf{b}<\/span><\/span>(<\/span>A<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span>A<\/span>)<\/span>(<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>b<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03b4<\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u4e14 \u2225\u03b4A\u2225<\u2225A\u22121\u2225\u22121\\\\|\\\\delta\\\\mathbf{A}\\\\| < \\\\|\\\\mathbf{A}^{-1}\\\\|^{-1}<\/span><\/span>\u2225<\/span>\u03b4<\/span>A<\/span>\u2225<\/span><\/span><<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span><\/span>\u2212<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u5219：
\n\u2225\u03b4x\u2225\u2225x\u2225\u2264\u03ba(A)1\u2212\u03ba(A)\u2225\u03b4A\u2225\u2225A\u2225(\u2225\u03b4A\u2225\u2225A\u2225+\u2225\u03b4b\u2225\u2225b\u2225)\\\\frac{\\\\|\\\\delta\\\\mathbf{x}\\\\|}{\\\\|\\\\mathbf{x}\\\\|} \\\\leq \\\\frac{\\\\kappa(\\\\mathbf{A})}{1 – \\\\kappa(\\\\mathbf{A})\\\\frac{\\\\|\\\\delta\\\\mathbf{A}\\\\|}{\\\\|\\\\mathbf{A}\\\\|}}\\\\left(\\\\frac{\\\\|\\\\delta\\\\mathbf{A}\\\\|}{\\\\|\\\\mathbf{A}\\\\|} + \\\\frac{\\\\|\\\\delta\\\\mathbf{b}\\\\|}{\\\\|\\\\mathbf{b}\\\\|}\\\\right)<\/span><\/span><\/span><\/span>\u2225<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>x<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>A<\/span>\u2225<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03ba<\/span>(<\/span>A<\/span>)<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span><\/span><\/span><\/span>\u2225<\/span>A<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>A<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>\u2225<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>\u03b4<\/span>b<\/span>\u2225<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u8868\u660e\u6761\u4ef6\u6570\u5927\u7684\u77e9\u9635\u5bf9\u6270\u52a8\u6781\u5176\u654f\u611f，\u8fd9\u662f\u6570\u503c\u8ba1\u7b97\u4e2d\u5fc5\u987b\u8003\u8651\u7684\u5173\u952e\u56e0\u7d20\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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