{"id":27,"date":"2014-12-11T14:40:03","date_gmt":"2014-12-11T14:40:03","guid":{"rendered":"http:\/\/c91.d5e.myftpupload.com\/?p=27"},"modified":"2015-01-01T14:45:21","modified_gmt":"2015-01-01T06:45:21","slug":"notes-on-conjugate-gradient-method","status":"publish","type":"post","link":"https:\/\/mindspectrum.xyz\/en\/2014\/12\/11\/notes-on-conjugate-gradient-method\/","title":{"rendered":"Notes on Conjugate Gradient Method"},"content":{"rendered":"
\u7248\u6743\u58f0\u660e\uff1a\u539f\u521b\u4f5c\u54c1\uff0c\u6b22\u8fce\u8f6c\u8f7d\uff0c\u4f46\u8f6c\u8f7d\u8bf7\u4ee5\u8d85\u94fe\u63a5\u5f62\u5f0f\u6ce8\u660e\u6587\u7ae0\u6765\u6e90(planckscale.info<\/a>)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002<\/strong><\/h5>\n

\u8fd9\u662f\u4e00\u4e2a\u5173\u4e8e\u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u7b14\u8bb0\u3002\u8bf7\u5927\u5bb6\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u662f\u4e2a\u7b14\u8bb0\uff0c\u5e76\u4e0d\u662f\u4e00\u4e2a\u6559\u7a0b\uff0c\u56e0\u6b64\u5c11\u4e0d\u4e86\u8df3\u8dc3\u548c\u6b20\u89e3\u91ca\u7684\u5730\u65b9\u3002\u5bf9CG\u65b9\u6cd5\u4e86\u89e3\u4e0d\u591a\u7684\u540c\u5b66\u8bf7\u79fb\u6b65\u8fd9\u91cc<\/a>\u3002<\/p>\n

\u7ebf\u6027\u65b9\u7a0b\u7ec4\u548c\u6781\u5c0f\u5316\u95ee\u9898<\/strong><\/h2>\n

\u4e00\u4e2a\u5173\u4e8e\u5bf9\u79f0\u77e9\u9635\"A\"\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\"Ax=b\"\u7b49\u4ef7\u4e8e\u6c42\u89e3\u5982\u4e0b\u6781\u5c0f\u503c\u95ee\u9898\uff1a<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

\u8fd9\u5f88\u5bb9\u6613\u8bf4\u660e\uff0c\u6211\u4eec\u5fae\u5206\u76ee\u6807\u51fd\u6570\u5f97<\/p>\n

(1) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u6240\u4ee5\"\td f=0\"\u610f\u5473\u7740\"Ax=b\".<\/p>\n

\u8bbe\"x^*\"\u4e3a\u95ee\u9898\u7684\u89e3\uff0c\"e\"\u4e3a\u504f\u79bb\u6781\u5c0f\u503c\u70b9\u7684\u4f4d\u79fb\uff0c\u5373<\/p>\n

(2) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u6211\u4eec\u5b9a\u4e49\u6b8b\u5dee\u6216\u8d1f\u68af\u5ea6\"r=b-Ax\"\uff0c\u5bb9\u6613\u770b\u51fa\uff0c\"r\"\u6b63\u6bd4\u4e8e\u504f\u79bb\u68af\u5ea6\u96f6\u70b9\"x^*\"\u7684\u4f4d\u79fb\"e\":<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

\u6700\u901f\u4e0b\u964d\u6cd5\u548c\u5171\u8f6d\u68af\u5ea6\u6cd5\u7b80\u8ff0<\/h2>\n

\u6700\u901f\u4e0b\u964d\u6cd5\uff1a\u641c\u7d22\u65b9\u5411\u4e3a\u672c\u8f6e\u8fed\u4ee3\u521d\u59cb\u70b9\u7684\u68af\u5ea6\u65b9\u5411\uff0c\u641c\u7d22\u5230\u68af\u5ea6\u4e0e\u641c\u7d22\u65b9\u5411\u6b63\u4ea4\u7684\u4f4d\u7f6e\u5f00\u59cb\u4e0b\u4e00\u8f6e\u8fed\u4ee3\u3002<\/p>\n

\u5171\u8f6d\u68af\u5ea6\u6cd5\uff1a\u641c\u7d22\u65b9\u5411\u4e3a\u672c\u8f6e\u8fed\u4ee3\u521d\u59cb\u70b9\u7684\u68af\u5ea6\u65b9\u5411\uff08\u6b8b\u5dee\u65b9\u5411\uff09\u7528A-\u5185\u79ef\u505a\u6b63\u4ea4\u5316\uff0c\u6263\u9664\u6389\u4e4b\u524d\u6240\u6709\u641c\u7d22\u65b9\u5411\u7684\u5206\u91cf\u7ed9\u51fa\uff0c\u641c\u7d22\u5230\u68af\u5ea6\u4e0e\u641c\u7d22\u65b9\u5411\u6b63\u4ea4\u7684\u4f4d\u7f6e\u5f00\u59cb\u4e0b\u4e00\u8f6e\u8fed\u4ee3\u3002<\/p>\n

\u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u7406\u89e3<\/h2>\n

\u9996\u5148\u8003\u8651\u4e00\u4e2a\u7b80\u5355\u60c5\u5f62\uff0c\u5373A=I\u65f6\u3002\u6b64\u65f6f\u9000\u5316\uff0c\u4e14x\u7684\u4e0d\u540c\u5206\u91cf\u89e3\u8026\uff0cf\u7684\u6781\u5c0f\u503c\u95ee\u9898\u5206\u89e3\u4e3a\u5404\u5206\u91cf\u4e0a\u7684\u6781\u5c0f\u503c\u95ee\u9898\u3002\u6211\u4eec\u53ea\u9700\u8981\u5728\u5404\u5206\u91cf\u65b9\u5411\u4e0a\u627e\u5230\u6781\u5c0f\u503c\uff0c\u95ee\u9898\u5c31\u5f97\u89e3\u4e86\u3002\u5177\u4f53\u7684\u8bf4\uff0c\u6211\u4eec\u4ece\u4efb\u610f\u4e00\u4e2a\u521d\u59cb\u70b9\u51fa\u53d1\uff0c\u5728\u65b9\u5411\"d_0\"\u4e0a\u5bfb\u627e\u6781\u5c0f\u503c\uff0c\u7136\u540e\u4ece\u8fd9\u4e2a\u6781\u5c0f\u503c\u51fa\u53d1\u7ee7\u7eed\u5728\"d_1\"\u5bfb\u627e\"d_1\"\u65b9\u5411\u7684\u6781\u5c0f\uff0c\u4ee5\u6b64\u7c7b\u63a8\u6700\u540e\u5230\u8fbe\u5168\u5c40\u6781\u5c0f\u503c\u3002<\/p>\n

A\u4e3a\u6b63\u5b9a\u5bf9\u79f0\u77e9\u9635\u7684\u4e00\u822c\u60c5\u5f62\uff0c\u5176\u5b9e\u53ea\u662f\u4e0a\u8ff0\u60c5\u5f62\u4e2d\u7684x\u8868\u8ff0\u5728\u4e00\u4e2a\u65b0\u5750\u6807\u7cfb\u4e0b\u7684\u7ed3\u679c\uff08\u4e0b\u6587\u8be6\u7ec6\u89e3\u91ca\uff09\u3002\u4e3a\u4e86\u6c42\u89e3\u8fd9\u79cd\u60c5\u5f62\uff0c\u6211\u4eec\u8003\u5bdf\u4e0a\u8ff0\u65b9\u6cd5\u7684\u6c42\u89e3\u8f68\u8ff9\u5728\u65b0\u5750\u6807\u7cfb\u91cc\u7684\u5bf9\u5e94\u3002<\/p>\n

\u9996\u5148\uff0cA=I\u65f6\u7684\u5404\u4e2a\u6b63\u4ea4\u641c\u7d22\u65b9\u5411\u5728\u65b0\u5750\u6807\u7cfb\u91cc\u5bf9\u5e94\u7740\u4e00\u7ec4A-\u6b63\u4ea4\u7684\u65b9\u5411\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u8bbe\u6cd5\u627e\u51fa\u8fd9\u6837\u4e00\u7ec4\u6b63\u4ea4\u65b9\u5411\uff0c\u7136\u540e\u5404\u81ea\u6c42\u89e3\u8fd9\u4e9b\u65b9\u5411\u4e0a\u7684\u6781\u5c0f\u503c\u3002<\/p>\n

\u6c42\u89e3\u7279\u5b9a\u65b9\u5411\u4e0a\u6781\u5c0f\u503c\u7684\u65b9\u6cd5\u4e0d\u53d8\uff0c\u6781\u5c0f\u70b9\u5c31\u662f\u8fd9\u4e2a\u65b9\u5411\u4e0a\u68af\u5ea6\u6b63\u4ea4\u4e8e\u641c\u7d22\u65b9\u5411\u65f6\u7684\u70b9\uff08\u539f\u56e0\u5f88\u7b80\u5355\uff0c\u8fd9\u8bf4\u660e\u641c\u7d22\u65b9\u5411\u4e0a\u68af\u5ea6\u5206\u91cf\u4e3a\u96f6\uff09\u3002<\/p>\n

\u6700\u540e\u4e00\u70b9\uff0c\u5982\u4f55\u627e\u5230\u8fd9\u6837\u4e00\u7ec4A-\u6b63\u4ea4\u65b9\u5411\uff1f<\/p>\n

\u4e00\u822c\u65b9\u6cd5\u662f\u627e\u4e00\u7ec4\u73b0\u6210\u7684\u5b8c\u5907\u57fa\uff0c\u8fdb\u884cA-\u6b63\u4ea4\u5316\u3002\u95ee\u9898\u662f\u8ba1\u7b97\u91cf\u6bd4\u8f83\u5927\u3002<\/p>\n

\u8bb0\u8fed\u4ee3\u5230\u7b2ci\u6b65\u65f6\uff0c\u6240\u641c\u7d22\u8fc7\u7684\u8fd9\u6837\u4e00\u7ec4A-\u6b63\u4ea4\u7684\u641c\u7d22\u65b9\u5411\u5355\u4f4d\u77e2\u91cf\u4e3a\"\{d_0,d_1,...,d_{i-1}\}\",\u5b83\u4eec\u6491\u8d77\u7684\u5b50\u7a7a\u95f4<\/p>\n

(3) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5373\u7b2c\"i\"\u6b65\u524d\u5df2\u7ecf\u641c\u7d22\u8fc7\u7684\u5b50\u7a7a\u95f4\u3002<\/p>\n

\u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u4e00\u4e2a\u5173\u952e\u4e4b\u5904\u5728\u4e8e\u5b83\u53ef\u4ee5\u4ece\u6bcf\u6b21\u8fed\u4ee3\u7684\"r_i\"\u4e2d\u5feb\u901f\u6784\u9020\u51fa\u65b0\u7684\u641c\u7d22\u65b9\u5411\"d_i\"\uff0c\u800c\u4e0d\u9700\u8981\u5bf9\"r_i\"\u8fdb\u884c\u5b8c\u6574\u7684\u6b63\u4ea4\u5316\u624b\u7eed\uff08\u5373\u4e0d\u9700\u4f9d\u6b21\u6263\u9664\"r_i\"\u4e2d\"d_0\",\"d_1\"…\"d_{i-1}\"\u65b9\u5411\u7684\u5206\u91cf\uff09\u3002\u539f\u56e0\u5728\u4e8e\"r_i\"\u81ea\u7136\u7684\u548c\"D_{i-1}\"A-\u6b63\u4ea4\uff0c\u56e0\u6b64\u6784\u9020\"d_i\"\u53ea\u9700\u8981\u4ece\"r_{i}\"\u4e2d\u6263\u9664\u6389\"d_{i-1}\"\u5206\u91cf\u5373\u53ef\u3002<\/p>\n

\u56e0\u6b64\uff0c\u5171\u8f6d\u68af\u5ea6\u7b97\u6cd5\u6709\u4e24\u5904\u5173\u952e\uff0c\u4e00\u5904\u4e3a\u901a\u8fc7\u5229\u7528A-\u6b63\u4ea4\u5173\u7cfb\u6765\u628a\u641c\u7d22\u95ee\u9898\u8f6c\u6362\u5230\u4e00\u4e2a\u7279\u6b8a\u5750\u6807\u7cfb\uff0c\u4f7f\u5f97\u5404\u641c\u7d22\u65b9\u5411\u4e0a\u7684\u6781\u5c0f\u503c\u95ee\u9898\u89e3\u8026\uff1b\u4e00\u662f\u5229\u7528\"r_i\"\u548c\"D_{i-1}\"\u662fA-\u6b63\u4ea4\u7684\u8fd9\u4e00\u89c4\u5f8b\u6765\u5145\u5206\u7b80\u5316A-\u6b63\u4ea4\u641c\u7d22\u65b9\u5411\u7684\u6784\u9020\u3002<\/strong><\/p>\n

\u63a5\u4e0b\u6765\u6211\u4eec\u8be6\u7ec6\u8bba\u8bc1\u8fd9\u4e2a\u7b97\u6cd5\u3002<\/p>\n

\u7b2c\u4e00\u4e2a\u5173\u952e\u70b9\uff0c\u5750\u6807\u53d8\u6362\u548c\u5b50\u95ee\u9898\u89e3\u8026<\/h2>\n

\u8bbe<\/p>\n

(4) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\"T^{-T}\"\u8868\u793a\u77e9\u9635\"T\"\u6c42\u9006\u540e\u505a\u8f6c\u7f6e\uff0c\u5f88\u62d7\u53e3\uff0c\u4e0d\u8fc7\u8fd9\u4e2a\u4e0d\u91cd\u8981\u3002<\/p>\n

\u4e8e\u662f\u6211\u4eec\u6709<\/p>\n

(5) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u53ef\u89c1\"f\"\u5df2\u7ecf\u88ab\u5206\u89e3\u4e3a\u4e00\u7cfb\u5217\u72ec\u7acb\u7684\u5b50\u95ee\u9898\"P_n(y_n)\"\uff0c\"T\"\u6b63\u662f\u5c06\u95ee\u9898\u89e3\u8026\u6240\u7528\u7684\u5750\u6807\u53d8\u6362\u3002<\/p>\n

\u5728\u65b0\u7684\"\{y_n\}\"\u5750\u6807\u7cfb\u4e2d\uff0c\u6574\u4e2a\u6781\u5c0f\u503c\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u5728\u4e00\u7ec4\u6b63\u4ea4\u65b9\u5411\u4e0a\u4f9d\u6b21\u5bfb\u627e\u6781\u5c0f\u70b9\u6765\u89e3\u51b3\u3002\u4f46\u6c42\u89e3\"T\"\u9700\u8981\u5bf9\"A\"\u505a\u5206\u89e3\uff0c\u5e76\u975e\u4e00\u4e2a\u8ba1\u7b97\u4e0a\u7ecf\u6d4e\u7684\u89e3\u51b3\u65b9\u6cd5\u3002\u8fd9\u4e00\u4e2a\u53d8\u6362\u7684\u771f\u6b63\u610f\u4e49\u5728\u4e8e\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u5b83\u6765\u627e\u5230\u6c42\u89e3\u8f68\u8ff9\u5728\"\{x_n\}\"\u5750\u6807\u7cfb\u4e2d\u7684\u5bf9\u5e94\u3002<\/p>\n

\"T\"\u662f\u7ebf\u6027\u53d8\u6362\uff0c\u56e0\u6b64\"\{y_n\}\"\u4e2d\u7684\u4e00\u7ec4\u76f4\u7ebf\u4ecd\u88ab\u53d8\u6362\u4e3a\"\{x_n\}\"\u4e2d\u7684\u4e00\u7ec4\u76f4\u7ebf\u3002\"\{y_n\}\"\u5750\u6807\u7cfb\u4e2d\uff0c\u6211\u4eec\u4f9d\u6b21\u6cbf\u7740\u4e00\u7ec4\u6b63\u4ea4\u65b9\u5411\u6c42\u89e3\u6781\u5c0f\u503c\uff0c\u56e0\u6b64\u53ea\u9700\u8981\u627e\u5230\u8fd9\u7ec4\u65b9\u5411\u5728\"\{x_n\}\"\u5750\u6807\u7cfb\u4e2d\u7684\u5bf9\u5e94\uff08\u4e00\u7ec4\u672a\u5fc5\u6b63\u4ea4\u7684\u65b9\u5411\uff09\uff0c\u5c31\u53ef\u4ee5\u7b49\u4ef7\u7684\u76f4\u63a5\u5728\u8fd9\u7ec4\u65b9\u5411\u4e4b\u4e0a\u5bfb\u627e\u6781\u5c0f\u503c\uff0c\u800c\u4e0d\u9700\u8981\u7ecf\u8fc7\u5750\u6807\u53d8\u6362\u3002<\/strong><\/p>\n

\u8003\u8651\u8fd9\u7ec4\u6b63\u4ea4\u65b9\u5411\u5355\u4f4d\u77e2\"\{y_n\}\"<\/p>\n

(6) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u53ef\u89c1\uff0c\"\{y_n\}\"\u8fd9\u4e00\u7ec4\u6b63\u4ea4\u7684\u65b9\u5411\u5728\u53d8\u6362\u5230\"\{x_n\}\"\u5750\u6807\u7cfb\u4e2d\u540e\uff0c\u867d\u7136\u4e0d\u662f\"\{x_n\}\"\u4e2d\u901a\u5e38\u610f\u4e49\u7684\u6b63\u4ea4\u77e2\u91cf\u7ec4\uff0c\u5374\u53ef\u4ee5\u5728A-\u5185\u79ef\"\langle \bullet,\bullet \rangle_A\"\u610f\u4e49\u4e0b\u6210\u4e3a\u4e00\u7ec4A-\u6b63\u4ea4\u77e2\u91cf\u3002<\/strong>\u8fd9\u91cc\u7684A-\u5185\u79ef\uff0c\u6307\u7684\u662f\u4ee5\u5bf9\u79f0\u77e9\u9635\"A\"\u5b9a\u4e49\u7684\u5185\u79ef\u8fd0\u7b97<\/p>\n

(7) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e0a\u8ff0\u7ed3\u8bba\u662f\u53ef\u9006\u7684\uff0c\u5982\u679c\"\langle x_i,x_j \rangle_A = 0\",\u540c\u6837\u4e5f\u53ef\u4ee5\u5f97\u51fa\uff0c\"x_i,x_j\"\u5bf9\u5e94\u7740\"\{y_n\}\"\u5750\u6807\u7cfb\u4e2d\u4e24\u4e2a\u6b63\u4ea4\u7684\u65b9\u5411\u3002\u56e0\u6b64\uff0c\u53ef\u4ee5\u76f4\u63a5\u5728\"\{x_n\}\"\u5750\u6807\u7cfb\u4e0b\uff0c\u5bfb\u627e\u4e00\u7ec4A-\u6b63\u4ea4\u7684\u65b9\u5411\uff0c\u7136\u540e\u5728\u8fd9\u4e00\u7ec4\u65b9\u5411\u4e0a\u4f9d\u6b21\u6781\u5c0f\u5316\u76ee\u6807\u51fd\u6570\uff0c\u5c31\u53ef\u4ee5\u627e\u5230\u6574\u4e2a\u95ee\u9898\u7684\u6781\u5c0f\u503c\u3002<\/p>\n

\u7b2c\u4e8c\u4e2a\u5173\u952e\u70b9\uff0c\"D_i\"\u5b50\u7a7a\u95f4\u7ed3\u6784\u548c\u6b63\u4ea4\u5173\u7cfb<\/h2>\n

\u7b97\u6cd5\u7684\u66f4\u65b0\u89c4\u5219\u4e3a<\/p>\n

(8) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5176\u4e2d\"\alpha_i\"\u4e3a\u7b2c\"i\"\u6b65\u641c\u7d22\u65b9\u5411\u4e0a\u7684\u4f4d\u79fb\u957f\u5ea6\uff0c\u5b83\u7684\u5927\u5c0f\u53ef\u4ee5\u901a\u8fc7\"r_{i+1}\"\u5e94\u8be5\u548c\"d_i\"\u6b63\u4ea4\u8fd9\u4e00\u8981\u6c42\u5b9a\u51fa<\/p>\n

(9) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u7b97\u6cd5\u5c06\u53c2\u6570\u7a7a\u95f4\u5212\u5206\u6210\u4e86\u4e00\u4e2a\u5c42\u7ea7\u7ed3\u6784\uff0c\u4e0b\u9762\u6211\u4eec\u6765\u770b\u770b\u8fd9\u4e2a\u7ed3\u6784\u7684\u6027\u8d28\u3002<\/p>\n

\u6839\u636e\u4e0a\u8ff0\u66f4\u65b0\u89c4\u5219\uff0c\u53ef\u4ee5\u53d1\u73b0\"D_i\"\u53ef\u4ee5\u7531\u53e6\u5916\u51e0\u4e2a\u77e2\u91cf\u7ec4\u6491\u8d77\u3002\u9996\u5148\uff0c\u7531\"d_i\"\u7684\u66f4\u65b0\u89c4\u5219\u53ef\u4ee5\u53d1\u73b0\uff0c\"d_i\"\u7531\"r_i\"\u548c\"d_{j<i}\"\u7ebf\u6027\u7ec4\u5408\u800c\u6210\uff0c\u53e6\u4e00\u65b9\u9762\uff0c\u6211\u4eec\u6709\u521d\u59cb\u6761\u4ef6\"d_0=r_0\"\uff0c\u5f52\u7eb3\u800c\u77e5\uff0c\"d_i\"\u53ef\u4ee5\u7531\"\{r_{j\leq i}\}\"\u7ebf\u6027\u7ec4\u5408\u51fa\u6765\u3002\u4e8e\u662f\u6211\u4eec\u6709<\/p>\n

(10) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u63a5\u7740\u6211\u4eec\u8003\u8651\"r_i\"\u7684\u66f4\u65b0\u89c4\u5219\u3002\u53ef\u4ee5\u770b\u5230\uff0c\"D_{i+1}=\mathrm{span}\{D_{i},Ad_i\}\".\u4e8e\u662f\u6211\u4eec\u53ef\u4ee5\u4ece\"D_1=\{d_0\}\"\u6216\"D_1=\{r_0\}\"\u6784\u9020\u51fa\"D_i\"<\/p>\n

(11) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u63a5\u4e0b\u6765\u6211\u4eec\u518d\u8003\u8651\u5404\u5b50\u7a7a\u95f4\u548c\u5411\u91cf\u4e4b\u95f4\u7684\u6b63\u4ea4\u5173\u7cfb\u3002\u9996\u5148\uff0c\"e_i\"\u5728\"D_i\"\u7684\u8865\u7a7a\u95f4\u91cc\u3002\u8fd9\u70b9\u662f\u660e\u663e\u7684\uff0c\u56e0\u4e3a\u6bcf\u4e00\u4e2a\u4f18\u5316\u6b65\u90fd\u4f1a\u4f18\u5316\u6389\"e_j\"\u5728\u76f8\u5e94\"d_j\"\u65b9\u5411\u4e0a\u7684\u5206\u91cf\uff0c\u56e0\u800c\u5269\u4f59\u7684\"e_i\"\u5c31\u53ea\u4f4d\u4e8e\"D_i\"\u7684\u8865\u7a7a\u95f4\u91cc\u3002\u4e8e\u662f\u6211\u4eec\u6709<\/p>\n

(12) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u8fd9\u6837\"e\"\u548c\"d\"\u4e4b\u95f4\u7684A-\u6b63\u4ea4\u5173\u7cfb\u5c31\u5f88\u660e\u663e\u4e86\u3002\u7531\u4e8e\"\langle d_i,d_j \rangle_A = \delta_{ij}\"\uff0c\u800c\"e_i\"\u53ea\u5305\u542b\"d_j>i\"\u5206\u91cf\uff0c\u56e0\u800c<\/p>\n

(13) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5373\"e_i\"\u548c\u5b50\u7a7a\u95f4\"D_i\"\u4e5f\u662fA-\u6b63\u4ea4\u7684\u3002\u8fd9\u610f\u5473\u7740\"r_i\"\u6b63\u4ea4\u4e8e\"D_i\"<\/p>\n

(14) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u5230\u8fd9\u91cc\u5c31\u53ef\u4ee5\u89e3\u91ca\u4e3a\u4ec0\u4e48\u6211\u4eec\u6709\"\langle r_i,d_{j<i-1} \rangle_A =0\"\u4e86\uff1a<\/p>\n

(15) <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

\u81f3\u6b64\u6211\u4eec\u5b8c\u6210\u4e86\u7b2c\u4e8c\u4e2a\u5173\u952e\u70b9\u7684\u8bba\u8bc1\uff0c\u4e14\u66f4\u6e05\u695a\u7684\u4e86\u89e3\u4e86\u53c2\u6570\u7a7a\u95f4\uff0c\u5c24\u5176\u662f\"D_i\"\u5b50\u7a7a\u95f4\u7684\u7ed3\u6784\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u7248\u6743\u58f0\u660e\uff1a\u539f\u521b\u4f5c\u54c1\uff0c\u6b22\u8fce\u8f6c\u8f7d\uff0c\u4f46\u8f6c\u8f7d\u8bf7\u4ee5\u8d85\u94fe\u63a5\u5f62\u5f0f\u6ce8\u660e\u6587\u7ae0\u6765\u6e90(planckscale.info)\u3001\u4f5c\u8005\u4fe1\u606f\u548c\u672c\u58f0\u660e\uff0c\u5426\u5219\u5c06\u8ffd\u7a76\u6cd5\u5f8b\u8d23\u4efb\u3002 \u8fd9\u662f\u4e00\u4e2a\u5173\u4e8e\u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u7b14\u8bb0\u3002\u8bf7\u5927\u5bb6\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u662f\u4e2a\u7b14\u8bb0\uff0c\u5e76\u4e0d\u662f\u4e00\u4e2a\u6559\u7a0b\uff0c\u56e0\u6b64\u5c11\u4e0d\u4e86\u8df3\u8dc3\u548c\u6b20\u89e3\u91ca\u7684\u5730\u65b9\u3002\u5bf9CG\u65b9\u6cd5\u4e86\u89e3\u4e0d\u591a\u7684\u540c\u5b66\u8bf7\u79fb\u6b65\u8fd9\u91cc\u3002 \u7ebf\u6027\u65b9\u7a0b\u7ec4\u548c\u6781\u5c0f\u5316\u95ee\u9898 \u4e00\u4e2a\u5173\u4e8e\u5bf9\u79f0\u77e9\u9635\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7b49\u4ef7\u4e8e\u6c42\u89e3\u5982\u4e0b\u6781\u5c0f\u503c\u95ee\u9898\uff1a     \u8fd9\u5f88\u5bb9\u6613\u8bf4\u660e\uff0c\u6211\u4eec\u5fae\u5206\u76ee\u6807\u51fd\u6570\u5f97 (1)   \u6240\u4ee5\u610f\u5473\u7740. \u8bbe\u4e3a\u95ee\u9898\u7684\u89e3\uff0c\u4e3a\u504f\u79bb\u6781\u5c0f\u503c\u70b9\u7684\u4f4d\u79fb\uff0c\u5373 (2)   \u6211\u4eec\u5b9a\u4e49\u6b8b\u5dee\u6216\u8d1f\u68af\u5ea6\uff0c\u5bb9\u6613\u770b\u51fa\uff0c\u6b63\u6bd4\u4e8e\u504f\u79bb\u68af\u5ea6\u96f6\u70b9\u7684\u4f4d\u79fb:     \u6700\u901f\u4e0b\u964d\u6cd5\u548c\u5171\u8f6d\u68af\u5ea6\u6cd5\u7b80\u8ff0 \u6700\u901f\u4e0b\u964d\u6cd5\uff1a\u641c\u7d22\u65b9\u5411\u4e3a\u672c\u8f6e\u8fed\u4ee3\u521d\u59cb\u70b9\u7684\u68af\u5ea6\u65b9\u5411\uff0c\u641c\u7d22\u5230\u68af\u5ea6\u4e0e\u641c\u7d22\u65b9\u5411\u6b63\u4ea4\u7684\u4f4d\u7f6e\u5f00\u59cb\u4e0b\u4e00\u8f6e\u8fed\u4ee3\u3002 \u5171\u8f6d\u68af\u5ea6\u6cd5\uff1a\u641c\u7d22\u65b9\u5411\u4e3a\u672c\u8f6e\u8fed\u4ee3\u521d\u59cb\u70b9\u7684\u68af\u5ea6\u65b9\u5411\uff08\u6b8b\u5dee\u65b9\u5411\uff09\u7528A-\u5185\u79ef\u505a\u6b63\u4ea4\u5316\uff0c\u6263\u9664\u6389\u4e4b\u524d\u6240\u6709\u641c\u7d22\u65b9\u5411\u7684\u5206\u91cf\u7ed9\u51fa\uff0c\u641c\u7d22\u5230\u68af\u5ea6\u4e0e\u641c\u7d22\u65b9\u5411\u6b63\u4ea4\u7684\u4f4d\u7f6e\u5f00\u59cb\u4e0b\u4e00\u8f6e\u8fed\u4ee3\u3002 \u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u7406\u89e3 \u9996\u5148\u8003\u8651\u4e00\u4e2a\u7b80\u5355\u60c5\u5f62\uff0c\u5373A=I\u65f6\u3002\u6b64\u65f6f\u9000\u5316\uff0c\u4e14x\u7684\u4e0d\u540c\u5206\u91cf\u89e3\u8026\uff0cf\u7684\u6781\u5c0f\u503c\u95ee\u9898\u5206\u89e3\u4e3a\u5404\u5206\u91cf\u4e0a\u7684\u6781\u5c0f\u503c\u95ee\u9898\u3002\u6211\u4eec\u53ea\u9700\u8981\u5728\u5404\u5206\u91cf\u65b9\u5411\u4e0a\u627e\u5230\u6781\u5c0f\u503c\uff0c\u95ee\u9898\u5c31\u5f97\u89e3\u4e86\u3002\u5177\u4f53\u7684\u8bf4\uff0c\u6211\u4eec\u4ece\u4efb\u610f\u4e00\u4e2a\u521d\u59cb\u70b9\u51fa\u53d1\uff0c\u5728\u65b9\u5411\u4e0a\u5bfb\u627e\u6781\u5c0f\u503c\uff0c\u7136\u540e\u4ece\u8fd9\u4e2a\u6781\u5c0f\u503c\u51fa\u53d1\u7ee7\u7eed\u5728\u5bfb\u627e\u65b9\u5411\u7684\u6781\u5c0f\uff0c\u4ee5\u6b64\u7c7b\u63a8\u6700\u540e\u5230\u8fbe\u5168\u5c40\u6781\u5c0f\u503c\u3002 A\u4e3a\u6b63\u5b9a\u5bf9\u79f0\u77e9\u9635\u7684\u4e00\u822c\u60c5\u5f62\uff0c\u5176\u5b9e\u53ea\u662f\u4e0a\u8ff0\u60c5\u5f62\u4e2d\u7684x\u8868\u8ff0\u5728\u4e00\u4e2a\u65b0\u5750\u6807\u7cfb\u4e0b\u7684\u7ed3\u679c\uff08\u4e0b\u6587\u8be6\u7ec6\u89e3\u91ca\uff09\u3002\u4e3a\u4e86\u6c42\u89e3\u8fd9\u79cd\u60c5\u5f62\uff0c\u6211\u4eec\u8003\u5bdf\u4e0a\u8ff0\u65b9\u6cd5\u7684\u6c42\u89e3\u8f68\u8ff9\u5728\u65b0\u5750\u6807\u7cfb\u91cc\u7684\u5bf9\u5e94\u3002 \u9996\u5148\uff0cA=I\u65f6\u7684\u5404\u4e2a\u6b63\u4ea4\u641c\u7d22\u65b9\u5411\u5728\u65b0\u5750\u6807\u7cfb\u91cc\u5bf9\u5e94\u7740\u4e00\u7ec4A-\u6b63\u4ea4\u7684\u65b9\u5411\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u8bbe\u6cd5\u627e\u51fa\u8fd9\u6837\u4e00\u7ec4\u6b63\u4ea4\u65b9\u5411\uff0c\u7136\u540e\u5404\u81ea\u6c42\u89e3\u8fd9\u4e9b\u65b9\u5411\u4e0a\u7684\u6781\u5c0f\u503c\u3002 \u6c42\u89e3\u7279\u5b9a\u65b9\u5411\u4e0a\u6781\u5c0f\u503c\u7684\u65b9\u6cd5\u4e0d\u53d8\uff0c\u6781\u5c0f\u70b9\u5c31\u662f\u8fd9\u4e2a\u65b9\u5411\u4e0a\u68af\u5ea6\u6b63\u4ea4\u4e8e\u641c\u7d22\u65b9\u5411\u65f6\u7684\u70b9\uff08\u539f\u56e0\u5f88\u7b80\u5355\uff0c\u8fd9\u8bf4\u660e\u641c\u7d22\u65b9\u5411\u4e0a\u68af\u5ea6\u5206\u91cf\u4e3a\u96f6\uff09\u3002 \u6700\u540e\u4e00\u70b9\uff0c\u5982\u4f55\u627e\u5230\u8fd9\u6837\u4e00\u7ec4A-\u6b63\u4ea4\u65b9\u5411\uff1f \u4e00\u822c\u65b9\u6cd5\u662f\u627e\u4e00\u7ec4\u73b0\u6210\u7684\u5b8c\u5907\u57fa\uff0c\u8fdb\u884cA-\u6b63\u4ea4\u5316\u3002\u95ee\u9898\u662f\u8ba1\u7b97\u91cf\u6bd4\u8f83\u5927\u3002 \u8bb0\u8fed\u4ee3\u5230\u7b2ci\u6b65\u65f6\uff0c\u6240\u641c\u7d22\u8fc7\u7684\u8fd9\u6837\u4e00\u7ec4A-\u6b63\u4ea4\u7684\u641c\u7d22\u65b9\u5411\u5355\u4f4d\u77e2\u91cf\u4e3a,\u5b83\u4eec\u6491\u8d77\u7684\u5b50\u7a7a\u95f4 (3)   \u5373\u7b2c\u6b65\u524d\u5df2\u7ecf\u641c\u7d22\u8fc7\u7684\u5b50\u7a7a\u95f4\u3002 \u5171\u8f6d\u68af\u5ea6\u6cd5\u7684\u4e00\u4e2a\u5173\u952e\u4e4b\u5904\u5728\u4e8e\u5b83\u53ef\u4ee5\u4ece\u6bcf\u6b21\u8fed\u4ee3\u7684\u4e2d\u5feb\u901f\u6784\u9020\u51fa\u65b0\u7684\u641c\u7d22\u65b9\u5411\uff0c\u800c\u4e0d\u9700\u8981\u5bf9\u8fdb\u884c\u5b8c\u6574\u7684\u6b63\u4ea4\u5316\u624b\u7eed\uff08\u5373\u4e0d\u9700\u4f9d\u6b21\u6263\u9664\u4e2d,…\u65b9\u5411\u7684\u5206\u91cf\uff09\u3002\u539f\u56e0\u5728\u4e8e\u81ea\u7136\u7684\u548cA-\u6b63\u4ea4\uff0c\u56e0\u6b64\u6784\u9020\u53ea\u9700\u8981\u4ece\u4e2d\u6263\u9664\u6389\u5206\u91cf\u5373\u53ef\u3002 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