{"id":292000,"date":"2024-05-20T14:07:29","date_gmt":"2024-05-20T06:07:29","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/292000.html"},"modified":"2024-05-20T14:07:35","modified_gmt":"2024-05-20T06:07:35","slug":"%e4%bb%80%e4%b9%88%e6%98%afg-s%e8%bf%ad%e4%bb%a3%e7%ae%97%e6%b3%95-2","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/292000.html","title":{"rendered":"\u4ec0\u4e48\u662fG-S\u8fed\u4ee3\u7b97\u6cd5"},"content":{"rendered":"

\"\u4ec0\u4e48\u662fG-S\u8fed\u4ee3\u7b97\u6cd5\"<\/p>\n

Gauss-Seidel\u8fed\u4ee3\u7b97\u6cd5\uff08G-S\u8fed\u4ee3\u7b97\u6cd5\uff09\u662f\u4e00\u79cd\u7528\u4e8e\u6c42\u89e3\u7ebf\u6027\u4ee3\u6570\u65b9\u7a0b\u7ec4\u7684\u8fed\u4ee3\u65b9\u6cd5\u3002\u5b83\u901a\u8fc7\u8fed\u4ee3\u6539\u8fdb\u89e3\u7684\u8fd1\u4f3c\u503c\uff0c\u76f4\u5230\u6ee1\u8db3\u4e00\u5b9a\u7684\u7cbe\u5ea6\u8981\u6c42\u4e3a\u6b62<\/strong>\u3002\u8be5\u65b9\u6cd5\u9002\u7528\u4e8e\u7cfb\u6570\u77e9\u9635\u4e3a\u5bf9\u89d2\u5360\u4f18\u6216\u6b63\u5b9a\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002G-S\u8fed\u4ee3\u7b97\u6cd5\u7684\u6838\u5fc3\u5728\u4e8e\u5229\u7528\u65b9\u7a0b\u7ec4\u4e2d\u5df2\u7ecf\u8ba1\u7b97\u51fa\u7684\u6700\u65b0\u89e3\u6765\u66f4\u65b0\u5f53\u524d\u6b63\u5728\u8ba1\u7b97\u7684\u89e3\uff0c\u4ece\u800c\u52a0\u901f\u6536\u655b\u8fc7\u7a0b\u3002\u6838\u5fc3\u7279\u70b9\u5305\u62ec\u8fed\u4ee3\u66f4\u65b0\u3001\u5c40\u90e8\u52a0\u901f\u548c\u4f9d\u8d56\u77e9\u9635\u7279\u6027<\/strong>\u3002<\/p>\n<\/p>\n

\u5c55\u5f00\u63cf\u8ff0\u5176\u8fed\u4ee3\u66f4\u65b0<\/strong>\u7279\u6027\uff1aG-S\u8fed\u4ee3\u7b97\u6cd5\u7684\u6bcf\u4e00\u6b21\u8fed\u4ee3\u66f4\u65b0\u90fd\u662f\u57fa\u4e8e\u5f53\u524d\u5df2\u6c42\u51fa\u7684\u6700\u65b0\u89e3\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5f53\u8ba1\u7b97\u65b9\u7a0b\u7ec4\u4e2d\u7684\u4e00\u4e2a\u672a\u77e5\u6570\u65f6\uff0c\u7b97\u6cd5\u4f1a\u5229\u7528\u8be5\u672a\u77e5\u6570\u4e4b\u524d\u7684\u65b9\u7a0b\u4e2d\u5df2\u7ecf\u8ba1\u7b97\u51fa\u7684\u6700\u65b0\u89e3\u3002\u8fd9\u79cd\u6309\u987a\u5e8f\u9010\u4e2a\u8ba1\u7b97\u5e76\u5373\u65f6\u5229\u7528\u6700\u65b0\u89e3\u66f4\u65b0\u7684\u65b9\u5f0f\uff0c\u4f7f\u5f97G-S\u8fed\u4ee3\u7b97\u6cd5\u76f8\u6bd4\u4e8e\u5176\u4ed6\u8fed\u4ee3\u7b97\u6cd5\uff08\u5982\u96c5\u53ef\u6bd4\u8fed\u4ee3\u6cd5\uff09\u80fd\u66f4\u5feb\u5730\u6536\u655b\u81f3\u6b63\u786e\u7b54\u6848\u3002<\/p>\n<\/p>\n

\u4e00\u3001G-S\u8fed\u4ee3\u7b97\u6cd5\u539f\u7406<\/h3>\n<\/p>\n

Gauss-Seidel\u8fed\u4ee3\u7b97\u6cd5\u57fa\u4e8e\u7ebf\u6027\u4ee3\u6570\u65b9\u7a0b\u7ec4\u7684\u77e9\u9635\u5f62\u5f0f\u3002\u8bbe\u6709\u7ebf\u6027\u65b9\u7a0b\u7ec4 (Ax = b)\uff0c\u5176\u4e2d\uff0c(A) \u662f\u4e00\u4e2a (n\\times n) \u7684\u7cfb\u6570\u77e9\u9635\uff0c(b) \u662f\u4e00\u4e2a (n\\times 1) \u7684\u5e38\u6570\u5411\u91cf\uff0c(x) \u662f\u9700\u8981\u6c42\u89e3\u7684\u672a\u77e5\u5411\u91cf\u3002G-S\u8fed\u4ee3\u7684\u57fa\u672c\u6b65\u9aa4\u662f\u5148\u5c06\u7cfb\u6570\u77e9\u9635 (A) \u5206\u89e3\u4e3a\u4e09\u90e8\u5206\uff1a\u5bf9\u89d2\u77e9\u9635 (D)\u3001\u4e0b\u4e09\u89d2\u77e9\u9635 (L) \u548c\u4e0a\u4e09\u89d2\u77e9\u9635 (U)\uff0c\u7136\u540e\u901a\u8fc7\u8c03\u6574\u8fd9\u4e9b\u5206\u91cf\u6765\u8fed\u4ee3\u6c42\u89e3 (x)\u3002<\/p>\n<\/p>\n

\u5728\u6bcf\u4e00\u6b21\u8fed\u4ee3\u4e2d\uff0c\u7b97\u6cd5\u8ba1\u7b97\u65b0\u7684 (x) \u503c\u4f5c\u4e3a\u8fd1\u4f3c\u89e3\uff0c\u901a\u8fc7\u4ea4\u66ff\u4f7f\u7528\u6700\u65b0\u8ba1\u7b97\u51fa\u7684\u503c\u66f4\u65b0\u5176\u4ed6\u503c\u3002\u8fd9\u79cd\u65b9\u6cd5\u7684\u6548\u7387\u5f88\u5927\u7a0b\u5ea6\u4e0a\u53d6\u51b3\u4e8e\u7cfb\u6570\u77e9\u9635\u7684\u7279\u6027\uff0c\u6bd4\u5982\u5bf9\u89d2\u5360\u4f18\u6216\u6b63\u5b9a\u6027\uff0c\u8fd9\u4e9b\u90fd\u4f1a\u5f71\u54cd\u5230\u7b97\u6cd5\u7684\u6536\u655b\u901f\u5ea6\u548c\u7a33\u5b9a\u6027\u3002<\/p>\n<\/p>\n

\u4e8c\u3001\u7b97\u6cd5\u6b65\u9aa4<\/h3>\n<\/p>\n

G-S\u8fed\u4ee3\u7b97\u6cd5\u7684\u5177\u4f53\u5b9e\u65bd\u6b65\u9aa4\u53ef\u4ee5\u5206\u4e3a\u51e0\u4e2a\u5173\u952e\u9636\u6bb5\uff1a<\/p>\n<\/p>\n

    \n
  1. \u521d\u59cb\u5316<\/strong>\uff1a\u9009\u62e9\u4e00\u4e2a\u521d\u59cb\u8fd1\u4f3c\u89e3 (x^{(0)})\uff0c\u8bbe\u7f6e\u8fed\u4ee3\u6b21\u6570\u4e3a0\u3002<\/li>\n
  2. \u8fed\u4ee3\u8ba1\u7b97<\/strong>\uff1a\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u672a\u77e5\u6570 (x_i)\uff0c\u5229\u7528\u6700\u65b0\u7684 (x_1, x_2, …, x_{i-1}) \u7684\u503c\u6309\u7167\u7279\u5b9a\u7684\u516c\u5f0f\u8fdb\u884c\u66f4\u65b0\u3002<\/li>\n
  3. \u6536\u655b\u5224\u65ad<\/strong>\uff1a\u8ba1\u7b97\u5f53\u524d\u8fed\u4ee3\u540e\u89e3\u7684\u8fd1\u4f3c\u503c\u4e0e\u4e0a\u4e00\u6b21\u8fed\u4ee3\u7684\u89e3\u4e4b\u95f4\u7684\u5dee\u5f02\u3002\u5982\u679c\u5dee\u5f02\u5c0f\u4e8e\u7528\u6237\u8bbe\u5b9a\u7684\u9608\u503c\uff0c\u5219\u8ba4\u4e3a\u5df2\u7ecf\u8fbe\u5230\u4e86\u9884\u671f\u7684\u7cbe\u5ea6\uff0c\u8fed\u4ee3\u8fc7\u7a0b\u7ed3\u675f\u3002\u5426\u5219\uff0c\u8fd4\u56de\u6b65\u9aa42\u7ee7\u7eed\u8fed\u4ee3\u3002<\/li>\n<\/ol>\n

    \u8fd9\u4e9b\u6b65\u9aa4\u5faa\u73af\u6267\u884c\uff0c\u76f4\u5230\u8fbe\u5230\u9884\u5b9a\u7684\u6536\u655b\u6761\u4ef6\u6216\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u3002<\/p>\n<\/p>\n

    \u4e09\u3001\u6536\u655b\u6761\u4ef6\u548c\u7a33\u5b9a\u6027<\/h3>\n<\/p>\n

    G-S\u8fed\u4ee3\u7b97\u6cd5\u7684\u6536\u655b\u6027\u662f\u5206\u6790\u7b97\u6cd5\u6548\u7387\u7684\u4e00\u4e2a\u91cd\u8981\u6307\u6807\u3002\u5bf9\u4e8e\u5bf9\u89d2\u5360\u4f18\u77e9\u9635\u6216\u6b63\u5b9a\u77e9\u9635\uff0cG-S\u8fed\u4ee3\u7b97\u6cd5\u901a\u5e38\u80fd\u4fdd\u8bc1\u6536\u655b\u6027<\/strong>\u3002\u5bf9\u89d2\u5360\u4f18\u662f\u6307\u77e9\u9635\u7684\u6bcf\u4e00\u884c\uff08\u6216\u6bcf\u4e00\u5217\uff09\u4e0a\u7684\u5bf9\u89d2\u5143\u7d20\u7684\u7edd\u5bf9\u503c\u5927\u4e8e\u8be5\u884c\uff08\u6216\u5217\uff09\u4e0a\u975e\u5bf9\u89d2\u5143\u7d20\u7edd\u5bf9\u503c\u4e4b\u548c\u3002\u6b63\u5b9a\u77e9\u9635\u5219\u662f\u6307\u6240\u6709\u7684\u7279\u5f81\u503c\u90fd\u4e3a\u6b63\u3002<\/p>\n<\/p>\n

    \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u68c0\u9a8c\u548c\u786e\u4fdd\u6536\u655b\u6027\u662f\u5b9e\u65bdG-S\u8fed\u4ee3\u7b97\u6cd5\u7684\u4e00\u4e2a\u91cd\u8981\u6b65\u9aa4\u3002\u4e0d\u540c\u7c7b\u578b\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u53ef\u80fd\u9700\u8981\u8c03\u6574\u8fed\u4ee3\u7b56\u7565\u6216\u8005\u5728\u7b97\u6cd5\u4e2d\u5f15\u5165\u677e\u5f1b\u56e0\u5b50\u6765\u63d0\u9ad8\u7a33\u5b9a\u6027\u548c\u6536\u655b\u901f\u5ea6\u3002<\/p>\n<\/p>\n

    \u56db\u3001\u5e94\u7528\u573a\u666f\u548c\u4f18\u52bf<\/h3>\n<\/p>\n

    G-S\u8fed\u4ee3\u7b97\u6cd5\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5de5\u7a0b\u3001\u7269\u7406\u5b66\u548c\u7ecf\u6d4e\u5b66\u4e2d\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u95ee\u9898\u3002\u5c24\u5176\u662f\u5728\u5904\u7406\u5927\u89c4\u6a21\u7a00\u758f\u77e9\u9635\u65f6\uff0c\u8be5\u7b97\u6cd5\u7531\u4e8e\u5176\u7b80\u5355\u548c\u4f4e\u5b58\u50a8\u9700\u6c42\u7684\u7279\u70b9\uff0c\u6210\u4e3a\u4e86\u4e00\u79cd\u975e\u5e38\u6709\u6548\u7684\u8ba1\u7b97\u5de5\u5177\u3002\u4f18\u52bf\u4e3b\u8981\u4f53\u73b0\u5728\uff1a\u4f4e\u5b58\u50a8\u9700\u6c42\u3001\u9002\u7528\u4e8e\u5927\u89c4\u6a21\u95ee\u9898\u548c\u9ad8\u6548\u6027<\/strong>\u3002<\/p>\n<\/p>\n

    \u5bf9\u4e8e\u4e00\u4e9b\u7279\u5b9a\u7c7b\u578b\u7684\u95ee\u9898\uff0c\u5982\u7a00\u758f\u77e9\u9635\u6216\u5bf9\u89d2\u5360\u4f18\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0cG-S\u8fed\u4ee3\u7b97\u6cd5\u4e0d\u4ec5\u80fd\u63d0\u4f9b\u7a33\u5b9a\u548c\u5feb\u901f\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u800c\u4e14\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\u6bd4\u4f20\u7edf\u7684\u6c42\u89e3\u65b9\u6cd5\u66f4\u4e3a\u9ad8\u6548\u3002<\/p>\n<\/p>\n

    \u603b\u4e4b\uff0cGauss-Seidel\u8fed\u4ee3\u7b97\u6cd5\u901a\u8fc7\u5176\u72ec\u7279\u7684\u8fed\u4ee3\u66f4\u65b0\u673a\u5236\u4ee5\u53ca\u5bf9\u7279\u5b9a\u7c7b\u578b\u77e9\u9635\u7684\u9002\u7528\u6027\uff0c\u5728\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u5c55\u73b0\u51fa\u4e86\u663e\u8457\u7684\u4f18\u52bf\u548c\u9ad8\u6548\u6027\u3002\u4e0d\u8fc7\uff0c\u5176\u6027\u80fd\u5f88\u5927\u7a0b\u5ea6\u4e0a\u4f9d\u8d56\u4e8e\u7cfb\u6570\u77e9\u9635\u7684\u7279\u6027\uff0c\u56e0\u6b64\uff0c\u5728\u5e94\u7528\u4e2d\u9700\u8981\u9488\u5bf9\u5177\u4f53\u95ee\u9898\u9009\u62e9\u5408\u9002\u7684\u8fed\u4ee3\u53c2\u6570\u4ee5\u786e\u4fdd\u7b97\u6cd5\u7684\u9ad8\u6548\u8fd0\u884c\u3002<\/p>\n<\/p>\n

    \u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n

    1. G-S\u8fed\u4ee3\u7b97\u6cd5\u662f\u4ec0\u4e48\uff1f<\/strong><\/p>\n

    G-S\u8fed\u4ee3\u7b97\u6cd5\u662f\u4e00\u79cd\u89e3\u51b3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8fed\u4ee3\u7b97\u6cd5\uff0c\u5b83\u7684\u5168\u79f0\u662fGauss-Seidel\u8fed\u4ee3\u7b97\u6cd5\u3002\u5b83\u662f\u57fa\u4e8e\u77e9\u9635\u5206\u89e3\u548c\u8fed\u4ee3\u7684\u601d\u60f3\uff0c\u901a\u8fc7\u4e0d\u65ad\u66f4\u65b0\u65b9\u7a0b\u7ec4\u7684\u89e3\u6765\u903c\u8fd1\u6700\u7ec8\u7684\u89e3\u3002<\/p>\n

    2. G-S\u8fed\u4ee3\u7b97\u6cd5\u6709\u4ec0\u4e48\u4f18\u52bf\u548c\u7f3a\u70b9\uff1f<\/strong><\/p>\n

    \u4f18\u52bf\u65b9\u9762\uff0cG-S\u8fed\u4ee3\u7b97\u6cd5\u4e0d\u9700\u8981\u6c42\u89e3\u6574\u4e2a\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u800c\u662f\u6309\u7167\u9010\u884c\u6216\u9010\u5217\u7684\u987a\u5e8f\u4e00\u6b21\u66f4\u65b0\u6bcf\u4e2a\u672a\u77e5\u53d8\u91cf\u7684\u503c\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u5e76\u884c\u6c42\u89e3\uff0c\u901f\u5ea6\u8f83\u5feb\u3002\u6b64\u5916\uff0cG-S\u7b97\u6cd5\u5bf9\u4e8e\u67d0\u4e9b\u7279\u6b8a\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u53ef\u80fd\u6bd4\u4f20\u7edf\u65b9\u6cd5\u66f4\u6536\u655b\u3002<\/p>\n

    \u7f3a\u70b9\u65b9\u9762\uff0cG-S\u8fed\u4ee3\u7b97\u6cd5\u5e76\u4e0d\u662f\u5bf9\u6240\u6709\u7ebf\u6027\u65b9\u7a0b\u7ec4\u90fd\u9002\u7528\uff0c\u5b83\u8981\u6c42\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u7cfb\u6570\u77e9\u9635\u662f\u5bf9\u89d2\u5360\u4f18\u7684\u3002\u53e6\u5916\uff0c\u5bf9\u4e8e\u6536\u655b\u901f\u5ea6\u8f83\u6162\u7684\u60c5\u51b5\uff0cG-S\u8fed\u4ee3\u7b97\u6cd5\u53ef\u80fd\u9700\u8981\u8fdb\u884c\u5927\u91cf\u7684\u8fed\u4ee3\u6b21\u6570\u624d\u80fd\u8fbe\u5230\u7cbe\u786e\u89e3\u3002<\/p>\n

    3. \u5982\u4f55\u4f7f\u7528G-S\u8fed\u4ee3\u7b97\u6cd5\u89e3\u51b3\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1f<\/strong><\/p>\n

    \u4f7f\u7528G-S\u8fed\u4ee3\u7b97\u6cd5\u89e3\u51b3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u4e00\u822c\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n

      \n
    • \u9996\u5148\uff0c\u5c06\u7ebf\u6027\u65b9\u7a0b\u7ec4\u8868\u793a\u4e3a\u7cfb\u6570\u77e9\u9635\u548c\u5e38\u6570\u5411\u91cf\u7684\u5f62\u5f0f\u3002<\/li>\n
    • \u7136\u540e\uff0c\u6839\u636eG-S\u8fed\u4ee3\u7b97\u6cd5\uff0c\u521d\u59cb\u5316\u672a\u77e5\u53d8\u91cf\u7684\u503c\u3002<\/li>\n
    • \u63a5\u7740\uff0c\u6309\u7167\u9010\u884c\u6216\u9010\u5217\u7684\u987a\u5e8f\uff0c\u66f4\u65b0\u5404\u4e2a\u672a\u77e5\u53d8\u91cf\u7684\u503c\uff0c\u76f4\u81f3\u8fbe\u5230\u6536\u655b\u6761\u4ef6\u3002<\/li>\n
    • \u6700\u540e\uff0c\u5f97\u5230\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8fd1\u4f3c\u89e3\u3002<\/li>\n<\/ul>\n

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