{"id":91416,"date":"2026-08-07T22:24:39","date_gmt":"2026-08-07T14:24:39","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/91416.html"},"modified":"2026-08-07T22:24:39","modified_gmt":"2026-08-07T14:24:39","slug":"%e4%bb%8e%e9%9b%b6%e5%ae%9e%e7%8e%b0%e7%9f%a9%e9%98%b5%e6%b1%82%e9%80%86%e4%b8%8e%e7%ba%bf%e6%80%a7%e6%96%b9%e7%a8%8b%e7%bb%84%e6%b1%82%e8%a7%a3%ef%bc%9a%e6%b7%b1%e5%85%a5%e7%90%86%e8%a7%a3%e9%ab%98","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/91416.html","title":{"rendered":"\u4ece\u96f6\u5b9e\u73b0\u77e9\u9635\u6c42\u9006\u4e0e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\uff1a\u6df1\u5165\u7406\u89e3\u9ad8\u65af\u6d88\u5143\u6cd5"},"content":{"rendered":"

1. \u5f15\u8a00<\/h3>\n

\u5728\u673a\u5668\u5b66\u4e60\u548c\u79d1\u5b66\u8ba1\u7b97\u4e2d，\u77e9\u9635\u6c42\u9006\u548c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u662f\u57fa\u7840\u800c\u91cd\u8981\u7684\u64cd\u4f5c\u3002\u867d\u7136 NumPy \u7b49\u5e93\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u5b9e\u73b0，\u4f46\u7406\u89e3\u5176\u5e95\u5c42\u539f\u7406\u5bf9\u4e8e\u6df1\u5165\u638c\u63e1\u7ebf\u6027\u4ee3\u6570\u81f3\u5173\u91cd\u8981\u3002\u672c\u6587\u5c06\u5e26\u4f60\u4ece\u96f6\u5b9e\u73b0\u77e9\u9635\u6c42\u9006\u548c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u7b97\u6cd5，\u901a\u8fc7\u4ee3\u7801\u5b9e\u8df5\u6df1\u5165\u7406\u89e3\u9ad8\u65af\u6d88\u5143\u6cd5\u7684\u539f\u7406\u3002<\/p>\n

2. \u77e9\u9635\u6c42\u9006\u7684\u5b9e\u73b0<\/h3>\n

2.1 \u7b97\u6cd5\u539f\u7406<\/h4>\n

\u77e9\u9635\u6c42\u9006\u7684\u6838\u5fc3\u601d\u60f3\u662f\u9ad8\u65af-\u7ea6\u65e6\u6d88\u5143\u6cd5\u3002\u5bf9\u4e8e\u4e00\u4e2a n\u00d7n \u7684\u65b9\u9635 A，\u6211\u4eec\u6784\u9020\u4e00\u4e2a\u589e\u5e7f\u77e9\u9635 [A|I]，\u5176\u4e2d I \u662f n\u00d7n \u7684\u5355\u4f4d\u77e9\u9635\u3002\u901a\u8fc7\u5bf9\u589e\u5e7f\u77e9\u9635\u8fdb\u884c\u884c\u53d8\u6362，\u5c06 A \u90e8\u5206\u5316\u4e3a\u5355\u4f4d\u77e9\u9635，\u6b64\u65f6\u53f3\u4fa7\u7684 I \u90e8\u5206\u5c31\u53d8\u6210\u4e86 A \u7684\u9006\u77e9\u9635\u3002<\/p>\n

2.2 \u4ee3\u7801\u5b9e\u73b0<\/h4>\n

import<\/span> numpy as<\/span> np<\/p>\n

def<\/span> matrix_inverse<\/span>(<\/span>a)<\/span>:<\/span>
\n """\u8ba1\u7b97\u65b9\u9635\u7684\u9006\u77e9\u9635"""<\/span>
\n n =<\/span> a.<\/span>shape[<\/span>0<\/span>]<\/span>
\n if<\/span> a.<\/span>shape[<\/span>0<\/span>]<\/span> !=<\/span> a.<\/span>shape[<\/span>1<\/span>]<\/span>:<\/span>
\n raise<\/span> ValueError(<\/span>"\u975e\u65b9\u9635，\u65e0\u6cd5\u8ba1\u7b97\u9006\u77e9\u9635\u3002"<\/span>)<\/span><\/p>\n

# \u6784\u9020\u589e\u5e7f\u77e9\u9635 [A|I]<\/span>
\n a_identity =<\/span> np.<\/span>hstack(<\/span>(<\/span>a,<\/span> np.<\/span>eye(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>)<\/span>)<\/span>)<\/span><\/p>\n

# \u8fdb\u884c\u9ad8\u65af-\u7ea6\u65e6\u6d88\u5143<\/span>
\n for<\/span> i in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
\n # \u5217\u4e3b\u5143\u9009\u62e9：\u627e\u5230\u7b2ci\u5217\u7edd\u5bf9\u503c\u6700\u5927\u7684\u884c<\/span>
\n max_row =<\/span> i
\n for<\/span> j in<\/span> range<\/span>(<\/span>i,<\/span> n)<\/span>:<\/span>
\n if<\/span> np.<\/span>abs<\/span>(<\/span>a_identity[<\/span>j,<\/span> i]<\/span>)<\/span> ><\/span> np.<\/span>abs<\/span>(<\/span>a_identity[<\/span>max_row,<\/span> i]<\/span>)<\/span>:<\/span>
\n max_row =<\/span> j<\/p>\n

# \u4ea4\u6362\u5f53\u524d\u884c\u4e0e\u4e3b\u5143\u884c<\/span>
\n if<\/span> max_row !=<\/span> i:<\/span>
\n a_identity[<\/span>[<\/span>i,<\/span> max_row]<\/span>,<\/span> :<\/span>]<\/span> =<\/span> a_identity[<\/span>[<\/span>max_row,<\/span> i]<\/span>,<\/span> :<\/span>]<\/span><\/p>\n

pivot =<\/span> a_identity[<\/span>i,<\/span> i]<\/span>
\n if<\/span> abs<\/span>(<\/span>pivot)<\/span> <<\/span> 1e-12<\/span>:<\/span>
\n raise<\/span> ValueError(<\/span>"\u77e9\u9635\u5947\u5f02，\u4e0d\u5b58\u5728\u9006\u77e9\u9635"<\/span>)<\/span><\/p>\n

# \u5c06\u4e3b\u5143\u884c\u5f52\u4e00\u5316<\/span>
\n a_identity[<\/span>i,<\/span> :<\/span>]<\/span> =<\/span> a_identity[<\/span>i,<\/span> :<\/span>]<\/span> \/<\/span> pivot<\/p>\n

# \u6d88\u53bb\u5176\u4ed6\u884c\u7684\u7b2ci\u5217\u5143\u7d20<\/span>
\n for<\/span> j in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
\n if<\/span> j !=<\/span> i:<\/span>
\n factor =<\/span> a_identity[<\/span>j,<\/span> i]<\/span>
\n a_identity[<\/span>j,<\/span> :<\/span>]<\/span> =<\/span> a_identity[<\/span>j,<\/span> :<\/span>]<\/span> –<\/span> a_identity[<\/span>i,<\/span> :<\/span>]<\/span> *<\/span> factor<\/p>\n

# \u8fd4\u56de\u9006\u77e9\u9635\u90e8\u5206<\/span>
\n return<\/span> a_identity[<\/span>:<\/span>,<\/span> n:<\/span>]<\/span><\/p>\n

# \u6d4b\u8bd5\u4ee3\u7801<\/span>
\nif<\/span> __name__ ==<\/span> "__main__"<\/span>:<\/span>
\n # \u6d4b\u8bd5\u77e9\u9635\u6c42\u9006<\/span>
\n a =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> [<\/span>4<\/span>,<\/span> 8<\/span>,<\/span> 6<\/span>]<\/span>,<\/span> [<\/span>7<\/span>,<\/span> 6<\/span>,<\/span> 9<\/span>]<\/span>]<\/span>)<\/span>
\n print<\/span>(<\/span>"\u539f\u59cb\u77e9\u9635 A:"<\/span>)<\/span>
\n print<\/span>(<\/span>a)<\/span>
\n print<\/span>(<\/span>"\\\\nNumPy \u8ba1\u7b97\u7684\u9006\u77e9\u9635:"<\/span>)<\/span>
\n print<\/span>(<\/span>np.<\/span>linalg.<\/span>inv(<\/span>a)<\/span>)<\/span>
\n print<\/span>(<\/span>"\\\\n\u81ea\u5b9a\u4e49\u5b9e\u73b0\u7684\u9006\u77e9\u9635:"<\/span>)<\/span>
\n custom_inv =<\/span> matrix_inverse(<\/span>a)<\/span>
\n print<\/span>(<\/span>custom_inv)<\/span><\/p>\n

# \u9a8c\u8bc1\u7ed3\u679c<\/span>
\n if<\/span> np.<\/span>allclose(<\/span>np.<\/span>linalg.<\/span>inv(<\/span>a)<\/span>,<\/span> custom_inv,<\/span> rtol=<\/span>1e-10<\/span>)<\/span>:<\/span>
\n print<\/span>(<\/span>"\\\\n\u2705 \u81ea\u5b9a\u4e49\u9006\u77e9\u9635\u8ba1\u7b97\u6b63\u786e"<\/span>)<\/span>
\n else<\/span>:<\/span>
\n print<\/span>(<\/span>"\\\\n\u274c \u81ea\u5b9a\u4e49\u9006\u77e9\u9635\u8ba1\u7b97\u9519\u8bef"<\/span>)<\/span><\/p>\n

2.3 \u5173\u952e\u4f18\u5316\u70b9<\/h4>\n
  • \u5217\u4e3b\u5143\u9009\u62e9：\u901a\u8fc7\u9009\u62e9\u7edd\u5bf9\u503c\u6700\u5927\u7684\u4e3b\u5143，\u63d0\u9ad8\u6570\u503c\u7a33\u5b9a\u6027<\/li>\n
  • \u77e9\u9635\u4ea4\u6362\u4f18\u5316：\u4f7f\u7528 NumPy \u7684\u9ad8\u7ea7\u7d22\u5f15\u8fdb\u884c\u884c\u4ea4\u6362，\u907f\u514d\u5faa\u73af<\/li>\n
  • \u5411\u91cf\u5316\u64cd\u4f5c：\u4f7f\u7528\u5e7f\u64ad\u673a\u5236\u66ff\u4ee3\u5185\u5c42\u5faa\u73af，\u63d0\u9ad8\u6548\u7387<\/li>\n
  • \u5947\u5f02\u77e9\u9635\u68c0\u6d4b：\u5f53\u4e3b\u5143\u63a5\u8fd1\u96f6\u65f6\u629b\u51fa\u5f02\u5e38<\/li>\n

    3. \u77e9\u9635\u4e58\u6cd5\u7684\u5b9e\u73b0<\/h3>\n

    3.1 \u57fa\u7840\u77e9\u9635\u4e58\u6cd5<\/h4>\n

    def<\/span> matrix_dot<\/span>(<\/span>a,<\/span> b)<\/span>:<\/span>
    \n """\u5b9e\u73b0\u77e9\u9635\u4e58\u6cd5，\u517c\u5bb9\u5411\u91cf\u8f93\u5165"""<\/span>
    \n # \u517c\u5bb9\u4e00\u7ef4\u5411\u91cf\u8f93\u5165<\/span>
    \n if<\/span> len<\/span>(<\/span>a.<\/span>shape)<\/span> ==<\/span> 1<\/span>:<\/span>
    \n a =<\/span> a.<\/span>reshape(<\/span>1<\/span>,<\/span> –<\/span>1<\/span>)<\/span>
    \n if<\/span> len<\/span>(<\/span>b.<\/span>shape)<\/span> ==<\/span> 1<\/span>:<\/span>
    \n b =<\/span> b.<\/span>reshape(<\/span>–<\/span>1<\/span>,<\/span> 1<\/span>)<\/span><\/p>\n

    # \u68c0\u67e5\u7ef4\u5ea6\u5339\u914d<\/span>
    \n ka =<\/span> a.<\/span>shape[<\/span>1<\/span>]<\/span>
    \n kb =<\/span> b.<\/span>shape[<\/span>0<\/span>]<\/span>
    \n if<\/span> ka !=<\/span> kb:<\/span>
    \n raise<\/span> ValueError(<\/span>f"\u77e9\u9635\u7ef4\u5ea6\u4e0d\u5339\u914d：a\u7684\u5217<\/span>{<\/span>ka}<\/span><\/span>，b\u7684\u884c<\/span>{<\/span>kb}<\/span><\/span>，\u65e0\u6cd5\u76f8\u4e58"<\/span><\/span>)<\/span><\/p>\n

    # \u4e09\u91cd\u5faa\u73af\u5b9e\u73b0\u77e9\u9635\u4e58\u6cd5<\/span>
    \n result =<\/span> np.<\/span>zeros(<\/span>(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>,<\/span> b.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>)<\/span>
    \n for<\/span> i in<\/span> range<\/span>(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>)<\/span>:<\/span>
    \n for<\/span> j in<\/span> range<\/span>(<\/span>b.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>
    \n s =<\/span> 0.0<\/span>
    \n for<\/span> k in<\/span> range<\/span>(<\/span>a.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>
    \n s +=<\/span> a[<\/span>i,<\/span> k]<\/span> *<\/span> b[<\/span>k,<\/span> j]<\/span>
    \n result[<\/span>i,<\/span> j]<\/span> =<\/span> s<\/p>\n

    return<\/span> result<\/p>\n

    # \u6d4b\u8bd5\u77e9\u9635\u4e58\u6cd5<\/span>
    \ndef<\/span> test_matrix_dot<\/span>(<\/span>)<\/span>:<\/span>
    \n a =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>]<\/span>,<\/span> [<\/span>3<\/span>,<\/span> 4<\/span>]<\/span>]<\/span>)<\/span>
    \n b =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>5<\/span>,<\/span> 6<\/span>]<\/span>,<\/span> [<\/span>7<\/span>,<\/span> 8<\/span>]<\/span>]<\/span>)<\/span><\/p>\n

    print<\/span>(<\/span>"\u77e9\u9635 A:"<\/span>)<\/span>
    \n print<\/span>(<\/span>a)<\/span>
    \n print<\/span>(<\/span>"\\\\n\u77e9\u9635 B:"<\/span>)<\/span>
    \n print<\/span>(<\/span>b)<\/span>
    \n print<\/span>(<\/span>"\\\\nNumPy \u8ba1\u7b97\u7ed3\u679c:"<\/span>)<\/span>
    \n print<\/span>(<\/span>np.<\/span>dot(<\/span>a,<\/span> b)<\/span>)<\/span>
    \n print<\/span>(<\/span>"\\\\n\u81ea\u5b9a\u4e49\u5b9e\u73b0\u7ed3\u679c:"<\/span>)<\/span>
    \n print<\/span>(<\/span>matrix_dot(<\/span>a,<\/span> b)<\/span>)<\/span><\/p>\n

    # \u9a8c\u8bc1\u5411\u91cf\u4e58\u6cd5<\/span>
    \n v1 =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>
    \n v2 =<\/span> np.<\/span>array(<\/span>[<\/span>4<\/span>,<\/span> 5<\/span>,<\/span> 6<\/span>]<\/span>)<\/span>
    \n print<\/span>(<\/span>"\\\\n\u5411\u91cf\u70b9\u79ef\u9a8c\u8bc1:"<\/span>)<\/span>
    \n print<\/span>(<\/span>"NumPy:"<\/span>,<\/span> np.<\/span>dot(<\/span>v1,<\/span> v2)<\/span>)<\/span>
    \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49:"<\/span>,<\/span> matrix_dot(<\/span>v1,<\/span> v2)<\/span>)<\/span><\/p>\n

    4. \u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3<\/h3>\n

    4.1 \u57fa\u4e8e\u9006\u77e9\u9635\u7684\u6c42\u89e3<\/h4>\n

    def<\/span> linear_equations_solve<\/span>(<\/span>a,<\/span> b)<\/span>:<\/span>
    \n """\u4f7f\u7528\u9006\u77e9\u9635\u6cd5\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax = b"""<\/span>
    \n # \u8bb0\u5f55\u539f\u59cbb\u662f\u5426\u4e3a\u4e00\u7ef4\u5411\u91cf<\/span>
    \n is_1d =<\/span> len<\/span>(<\/span>b.<\/span>shape)<\/span> ==<\/span> 1<\/span><\/p>\n

    # \u8ba1\u7b97\u9006\u77e9\u9635<\/span>
    \n inverse_a =<\/span> matrix_inverse(<\/span>a)<\/span><\/p>\n

    # \u8ba1\u7b97\u89e3：x = A\u207b\u00b9b<\/span>
    \n res =<\/span> matrix_dot(<\/span>inverse_a,<\/span> b)<\/span><\/p>\n

    # \u5982\u679c\u539f\u59cbb\u662f\u4e00\u7ef4\u5411\u91cf，\u5c06\u7ed3\u679c\u8f6c\u6362\u4e3a\u4e00\u7ef4<\/span>
    \n if<\/span> is_1d:<\/span>
    \n res =<\/span> res.<\/span>flatten(<\/span>)<\/span><\/p>\n

    return<\/span> res<\/p>\n

    # \u6d4b\u8bd5\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3<\/span>
    \ndef<\/span> test_linear_solve<\/span>(<\/span>)<\/span>:<\/span>
    \n # \u6d4b\u8bd5\u7528\u4f8b1：\u4e00\u822c\u77e9\u9635<\/span>
    \n a =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> [<\/span>4<\/span>,<\/span> 8<\/span>,<\/span> 6<\/span>]<\/span>,<\/span> [<\/span>7<\/span>,<\/span> 6<\/span>,<\/span> 9<\/span>]<\/span>]<\/span>)<\/span>
    \n b =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>)<\/span><\/p>\n

    print<\/span>(<\/span>"\u6d4b\u8bd5\u7528\u4f8b1 – \u4e00\u822c\u77e9\u9635"<\/span>)<\/span>
    \n print<\/span>(<\/span>"\u7cfb\u6570\u77e9\u9635 A:"<\/span>)<\/span>
    \n print<\/span>(<\/span>a)<\/span>
    \n print<\/span>(<\/span>"\\\\n\u5e38\u6570\u5411\u91cf b:"<\/span>,<\/span> b)<\/span><\/p>\n

    print<\/span>(<\/span>"\\\\nNumPy \u6c42\u89e3\u7ed3\u679c:"<\/span>)<\/span>
    \n np_solution =<\/span> np.<\/span>linalg.<\/span>solve(<\/span>a,<\/span> b)<\/span>
    \n print<\/span>(<\/span>np_solution)<\/span><\/p>\n

    print<\/span>(<\/span>"\\\\n\u81ea\u5b9a\u4e49\u6c42\u89e3\u7ed3\u679c:"<\/span>)<\/span>
    \n custom_solution =<\/span> linear_equations_solve(<\/span>a,<\/span> b)<\/span>
    \n print<\/span>(<\/span>custom_solution)<\/span><\/p>\n

    if<\/span> np.<\/span>allclose(<\/span>np_solution,<\/span> custom_solution,<\/span> rtol=<\/span>1e-10<\/span>)<\/span>:<\/span>
    \n print<\/span>(<\/span>"\\\\n\u2705 \u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u6b63\u786e"<\/span>)<\/span>
    \n else<\/span>:<\/span>
    \n print<\/span>(<\/span>"\\\\n\u274c \u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u9519\u8bef"<\/span>)<\/span><\/p>\n

    # \u6d4b\u8bd5\u7528\u4f8b2：\u5355\u4f4d\u77e9\u9635<\/span>
    \n print<\/span>(<\/span>"\\\\n\\\\n\u6d4b\u8bd5\u7528\u4f8b2 – \u5355\u4f4d\u77e9\u9635"<\/span>)<\/span>
    \n a_identity =<\/span> np.<\/span>eye(<\/span>3<\/span>)<\/span>
    \n b_vec =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>)<\/span><\/p>\n

    print<\/span>(<\/span>"\u7cfb\u6570\u77e9\u9635 A (\u5355\u4f4d\u77e9\u9635):"<\/span>)<\/span>
    \n print<\/span>(<\/span>a_identity)<\/span>
    \n print<\/span>(<\/span>"\\\\n\u5e38\u6570\u5411\u91cf b:"<\/span>,<\/span> b_vec)<\/span><\/p>\n

    print<\/span>(<\/span>"\\\\nNumPy \u6c42\u89e3\u7ed3\u679c:"<\/span>)<\/span>
    \n print<\/span>(<\/span>np.<\/span>linalg.<\/span>solve(<\/span>a_identity,<\/span> b_vec)<\/span>)<\/span><\/p>\n

    print<\/span>(<\/span>"\\\\n\u81ea\u5b9a\u4e49\u6c42\u89e3\u7ed3\u679c:"<\/span>)<\/span>
    \n print<\/span>(<\/span>linear_equations_solve(<\/span>a_identity,<\/span> b_vec)<\/span>)<\/span><\/p>\n

    5. \u7b97\u6cd5\u590d\u6742\u5ea6\u5206\u6790<\/h3>\n

    5.1 \u65f6\u95f4\u590d\u6742\u5ea6<\/h4>\n\n\u64cd\u4f5c\u65f6\u95f4\u590d\u6742\u5ea6\u8bf4\u660e<\/tr>\n\n\n\n\n
    \u77e9\u9635\u6c42\u9006<\/td>\nO(n\u00b3)<\/td>\n\u9ad8\u65af\u6d88\u5143\u6cd5\u7684\u6807\u51c6\u590d\u6742\u5ea6<\/td>\n<\/tr>\n
    \u77e9\u9635\u4e58\u6cd5<\/td>\nO(n\u00b3)<\/td>\n\u4e09\u91cd\u5faa\u73af\u5b9e\u73b0<\/td>\n<\/tr>\n
    \u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3<\/td>\nO(n\u00b3)<\/td>\n\u4e3b\u8981\u5f00\u9500\u5728\u6c42\u9006<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    5.2 \u7a7a\u95f4\u590d\u6742\u5ea6<\/h4>\n
      \n
    • \u77e9\u9635\u6c42\u9006：O(n\u00b2)，\u9700\u8981\u5b58\u50a8\u589e\u5e7f\u77e9\u9635<\/li>\n
    • \u77e9\u9635\u4e58\u6cd5：O(n\u00b2)，\u9700\u8981\u5b58\u50a8\u7ed3\u679c\u77e9\u9635<\/li>\n
    • \u6574\u4f53\u7b97\u6cd5：O(n\u00b2)<\/li>\n<\/ul>\n

      6. \u6570\u503c\u7a33\u5b9a\u6027\u8003\u8651<\/h3>\n

      6.1 \u4e3b\u5143\u9009\u62e9\u7684\u91cd\u8981\u6027<\/h4>\n

      def<\/span> test_pivot_importance<\/span>(<\/span>)<\/span>:<\/span>
      \n """\u6f14\u793a\u4e3b\u5143\u9009\u62e9\u5bf9\u6570\u503c\u7a33\u5b9a\u6027\u7684\u5f71\u54cd"""<\/span>
      \n # \u521b\u5efa\u4e00\u4e2a\u6761\u4ef6\u6570\u8f83\u5927\u7684\u77e9\u9635<\/span>
      \n a =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>1e-10<\/span>,<\/span> 1<\/span>]<\/span>,<\/span> [<\/span>1<\/span>,<\/span> 1<\/span>]<\/span>]<\/span>)<\/span>
      \n b =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>]<\/span>)<\/span><\/p>\n

      print<\/span>(<\/span>"\u6761\u4ef6\u6570\u8f83\u5927\u7684\u77e9\u9635:"<\/span>)<\/span>
      \n print<\/span>(<\/span>"A ="<\/span>,<\/span> a)<\/span>
      \n print<\/span>(<\/span>"b ="<\/span>,<\/span> b)<\/span><\/p>\n

      try<\/span>:<\/span>
      \n # \u4e0d\u4f7f\u7528\u4e3b\u5143\u9009\u62e9（\u7b80\u5316\u7248\u672c）<\/span>
      \n print<\/span>(<\/span>"\\\\n\u4e0d\u4f7f\u7528\u4e3b\u5143\u9009\u62e9\u7684\u7ed3\u679c:"<\/span>)<\/span>
      \n # \u8fd9\u91cc\u53ef\u4ee5\u5bf9\u6bd4\u4e0d\u540c\u5b9e\u73b0<\/span>
      \n except<\/span> Exception as<\/span> e:<\/span>
      \n print<\/span>(<\/span>f"\u8ba1\u7b97\u5931\u8d25: <\/span>{<\/span>e}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

      print<\/span>(<\/span>"\\\\n\u4f7f\u7528\u5217\u4e3b\u5143\u9009\u62e9\u7684\u7ed3\u679c:"<\/span>)<\/span>
      \n solution =<\/span> linear_equations_solve(<\/span>a,<\/span> b)<\/span>
      \n print<\/span>(<\/span>"x ="<\/span>,<\/span> solution)<\/span><\/p>\n

      6.2 \u6d6e\u70b9\u6570\u7cbe\u5ea6\u5904\u7406<\/h4>\n

      \u5728\u5b9e\u73b0\u4e2d\u9700\u8981\u6ce8\u610f：<\/p>\n

    • \u4f7f\u7528 abs(pivot) < 1e-12 \u5224\u65ad\u5947\u5f02\u77e9\u9635<\/li>\n
    • \u4f7f\u7528 np.allclose() \u6bd4\u8f83\u6d6e\u70b9\u6570\u7ed3\u679c<\/li>\n
    • \u907f\u514d\u7d2f\u79ef\u8bef\u5dee<\/li>\n

      7. \u6027\u80fd\u4f18\u5316\u5efa\u8bae<\/h3>\n

      7.1 \u5411\u91cf\u5316\u4f18\u5316<\/h4>\n

      def<\/span> matrix_dot_optimized<\/span>(<\/span>a,<\/span> b)<\/span>:<\/span>
      \n """\u4f18\u5316\u7248\u7684\u77e9\u9635\u4e58\u6cd5"""<\/span>
      \n if<\/span> len<\/span>(<\/span>a.<\/span>shape)<\/span> ==<\/span> 1<\/span>:<\/span>
      \n a =<\/span> a.<\/span>reshape(<\/span>1<\/span>,<\/span> –<\/span>1<\/span>)<\/span>
      \n if<\/span> len<\/span>(<\/span>b.<\/span>shape)<\/span> ==<\/span> 1<\/span>:<\/span>
      \n b =<\/span> b.<\/span>reshape(<\/span>–<\/span>1<\/span>,<\/span> 1<\/span>)<\/span><\/p>\n

      # \u4f7f\u7528 @ \u8fd0\u7b97\u7b26\u6216 np.dot \u7684\u4f18\u5316\u7248\u672c<\/span>
      \n return<\/span> a @ b # \u6216\u8005 np.dot(a, b)<\/span><\/p>\n

      def<\/span> matrix_inverse_optimized<\/span>(<\/span>a)<\/span>:<\/span>
      \n """\u4f18\u5316\u7248\u7684\u77e9\u9635\u6c42\u9006"""<\/span>
      \n n =<\/span> a.<\/span>shape[<\/span>0<\/span>]<\/span>
      \n a_identity =<\/span> np.<\/span>hstack(<\/span>(<\/span>a,<\/span> np.<\/span>eye(<\/span>n)<\/span>)<\/span>)<\/span><\/p>\n

      for<\/span> i in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
      \n # \u5217\u4e3b\u5143\u9009\u62e9（\u5411\u91cf\u5316）<\/span>
      \n max_row =<\/span> i +<\/span> np.<\/span>argmax(<\/span>np.<\/span>abs<\/span>(<\/span>a_identity[<\/span>i:<\/span>,<\/span> i]<\/span>)<\/span>)<\/span>
      \n if<\/span> max_row !=<\/span> i:<\/span>
      \n a_identity[<\/span>[<\/span>i,<\/span> max_row]<\/span>]<\/span> =<\/span> a_identity[<\/span>[<\/span>max_row,<\/span> i]<\/span>]<\/span><\/p>\n

      pivot =<\/span> a_identity[<\/span>i,<\/span> i]<\/span>
      \n if<\/span> abs<\/span>(<\/span>pivot)<\/span> <<\/span> 1e-12<\/span>:<\/span>
      \n raise<\/span> ValueError(<\/span>"\u77e9\u9635\u5947\u5f02"<\/span>)<\/span><\/p>\n

      # \u5f52\u4e00\u5316\u5f53\u524d\u884c<\/span>
      \n a_identity[<\/span>i]<\/span> \/=<\/span> pivot<\/p>\n

      # \u6d88\u53bb\u5176\u4ed6\u884c（\u5411\u91cf\u5316）<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
      \n if<\/span> j !=<\/span> i:<\/span>
      \n a_identity[<\/span>j]<\/span> -=<\/span> a_identity[<\/span>i]<\/span> *<\/span> a_identity[<\/span>j,<\/span> i]<\/span><\/p>\n

      return<\/span> a_identity[<\/span>:<\/span>,<\/span> n:<\/span>]<\/span><\/p>\n

      7.2 \u4f7f\u7528 BLAS\/LAPACK<\/h4>\n

      \u5bf9\u4e8e\u751f\u4ea7\u73af\u5883，\u5efa\u8bae：<\/p>\n

    • \u4f7f\u7528 NumPy \u7684\u5e95\u5c42 BLAS\/LAPACK \u5b9e\u73b0<\/li>\n
    • \u8003\u8651\u4f7f\u7528 SciPy \u7684\u4f18\u5316\u7b97\u6cd5<\/li>\n
    • \u5bf9\u4e8e\u5927\u578b\u77e9\u9635，\u4f7f\u7528\u8fed\u4ee3\u6cd5\u6216\u5206\u89e3\u6cd5<\/li>\n

      8. \u603b\u7ed3<\/h3>\n

      \u672c\u6587\u4ece\u96f6\u5b9e\u73b0\u4e86\u77e9\u9635\u6c42\u9006\u548c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u7684\u6838\u5fc3\u7b97\u6cd5，\u91cd\u70b9\u5305\u62ec：<\/p>\n

    • \u9ad8\u65af-\u7ea6\u65e6\u6d88\u5143\u6cd5：\u901a\u8fc7\u589e\u5e7f\u77e9\u9635\u53d8\u6362\u5b9e\u73b0\u77e9\u9635\u6c42\u9006<\/li>\n
    • \u5217\u4e3b\u5143\u9009\u62e9：\u63d0\u9ad8\u6570\u503c\u7a33\u5b9a\u6027<\/li>\n
    • \u77e9\u9635\u4e58\u6cd5：\u57fa\u7840\u7684\u4e09\u91cd\u5faa\u73af\u5b9e\u73b0<\/li>\n
    • \u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3：\u57fa\u4e8e\u9006\u77e9\u9635\u7684\u65b9\u6cd5<\/li>\n

      \u867d\u7136\u8fd9\u4e9b\u5b9e\u73b0\u4e0d\u5982 NumPy \u4f18\u5316，\u4f46\u7406\u89e3\u5176\u539f\u7406\u5bf9\u4e8e\u6df1\u5165\u5b66\u4e60\u7ebf\u6027\u4ee3\u6570\u548c\u6570\u503c\u8ba1\u7b97\u81f3\u5173\u91cd\u8981\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d，\u5efa\u8bae\u4f7f\u7528\u7ecf\u8fc7\u9ad8\u5ea6\u4f18\u5316\u7684\u6570\u503c\u8ba1\u7b97\u5e93，\u4f46\u5728\u5b66\u4e60\u548c\u6559\u5b66\u573a\u666f\u4e2d，\u624b\u52a8\u5b9e\u73b0\u8fd9\u4e9b\u7b97\u6cd5\u662f\u6781\u597d\u7684\u7ec3\u4e60\u3002<\/p>\n

      9. \u6269\u5c55\u7ec3\u4e60<\/h3>\n
    • \u5b9e\u73b0 LU \u5206\u89e3\u6cd5\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/li>\n
    • \u5b9e\u73b0 QR \u5206\u89e3\u6cd5\u6c42\u9006\u77e9\u9635<\/li>\n
    • \u6dfb\u52a0\u5e76\u884c\u8ba1\u7b97\u4f18\u5316<\/li>\n
    • \u5b9e\u73b0\u7a00\u758f\u77e9\u9635\u7684\u4f18\u5316\u7248\u672c<\/li>\n
    • \u6dfb\u52a0\u8fed\u4ee3\u7cbe\u5316（Iterative Refinement）\u63d0\u9ad8\u7cbe\u5ea6<\/li>\n

      # \u6269\u5c55：\u9a8c\u8bc1\u9006\u77e9\u9635\u6027\u8d28<\/span>
      \ndef<\/span> verify_inverse_properties<\/span>(<\/span>a)<\/span>:<\/span>
      \n """\u9a8c\u8bc1\u9006\u77e9\u9635\u7684\u57fa\u672c\u6027\u8d28"""<\/span>
      \n inv_a =<\/span> matrix_inverse(<\/span>a)<\/span><\/p>\n

      # \u6027\u8d281: A \u00d7 A\u207b\u00b9 = I<\/span>
      \n product1 =<\/span> matrix_dot(<\/span>a,<\/span> inv_a)<\/span>
      \n identity1 =<\/span> np.<\/span>eye(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>)<\/span><\/p>\n

      # \u6027\u8d282: A\u207b\u00b9 \u00d7 A = I<\/span>
      \n product2 =<\/span> matrix_dot(<\/span>inv_a,<\/span> a)<\/span>
      \n identity2 =<\/span> np.<\/span>eye(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>)<\/span><\/p>\n

      print<\/span>(<\/span>f"A \u00d7 A\u207b\u00b9 \u63a5\u8fd1\u5355\u4f4d\u77e9\u9635: <\/span>{<\/span>np.<\/span>allclose(<\/span>product1,<\/span> identity1)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
      \n print<\/span>(<\/span>f"A\u207b\u00b9 \u00d7 A \u63a5\u8fd1\u5355\u4f4d\u77e9\u9635: <\/span>{<\/span>np.<\/span>allclose(<\/span>product2,<\/span> identity2)<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

      return<\/span> inv_a<\/p>\n

      \u901a\u8fc7\u672c\u6587\u7684\u5b66\u4e60，\u4f60\u5e94\u8be5\u5bf9\u77e9\u9635\u6c42\u9006\u548c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u6709\u4e86\u66f4\u6df1\u5165\u7684\u7406\u89e3\u3002\u7ee7\u7eed\u63a2\u7d22\u6570\u503c\u7ebf\u6027\u4ee3\u6570\u7684\u4e16\u754c，\u4f60\u4f1a\u53d1\u73b0\u66f4\u591a\u6709\u8da3\u7684\u7b97\u6cd5\u548c\u5e94\u7528！<\/p>\n

      \u9644<\/h2>\n

      \u4ee5\u4e0a\u4e3aAI\u57fa\u4e8e\u4ee5\u4e0b\u6d4b\u8bd5\u9a8c\u8bc1\u4ee3\u7801\u751f\u6210，\u53ef\u4ee5\u4f5c\u4e3a\u53c2\u8003\u3002<\/p>\n

      import<\/span> numpy as<\/span> np<\/p>\n

      def<\/span> matrix_inverse<\/span>(<\/span>a)<\/span>:<\/span>
      \n n =<\/span> a.<\/span>shape[<\/span>0<\/span>]<\/span>
      \n if<\/span> a.<\/span>shape[<\/span>0<\/span>]<\/span> !=<\/span> a.<\/span>shape[<\/span>1<\/span>]<\/span>:<\/span>
      \n raise<\/span> ValueError(<\/span>"\u975e\u65b9\u9635，\u65e0\u6cd5\u8ba1\u7b97\u9006\u77e9\u9635\u3002"<\/span>)<\/span><\/p>\n

      a_identity =<\/span> np.<\/span>hstack(<\/span>(<\/span>a,<\/span> np.<\/span>eye(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>)<\/span>)<\/span>)<\/span>
      \n #\u5148\u628a\u6269\u5c55\u4e3an*2n\u7684\u77e9\u9635<\/span>
      \n # a_identity = np.zeros((n, 2*n), dtype=float)<\/span>
      \n # a_identity[:, :n] = a<\/span>
      \n # #\u5148\u62fc\u63a5\u57fa\u7840\u77e9\u9635<\/span>
      \n # for i in range(n):<\/span>
      \n # for j in range(n, a_identity.shape[1]):<\/span>
      \n # if i == j – n:<\/span>
      \n # a_identity[i, j] = 1<\/span>
      \n # else:<\/span>
      \n # a_identity[i, j] = 0<\/span>
      \n # \u8fdb\u884c\u9ad8\u65af\u6d88\u5143\u6cd5<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
      \n #\u627e\u5230\u672c\u5217\u7edd\u5bf9\u503c\u6700\u5927\u7684\u884c<\/span>
      \n max_row =<\/span> i
      \n for<\/span> j in<\/span> range<\/span>(<\/span>i,<\/span> n)<\/span>:<\/span> # \u4ecei\u884c\u5411\u4e0b\u5f00\u59cb\u67e5\u627e\u7edd\u5bf9\u503c\u6700\u5927\u7684\u884c，\u4e0a\u9762\u7684\u662f\u5df2\u7ecf\u4ea4\u6362\u8fc7\u7684<\/span>
      \n if<\/span> np.<\/span>abs<\/span>(<\/span>a_identity[<\/span>j,<\/span> i]<\/span>)<\/span> ><\/span> np.<\/span>abs<\/span>(<\/span>a_identity[<\/span>max_row,<\/span> i]<\/span>)<\/span>:<\/span>
      \n max_row =<\/span> j
      \n temp =<\/span> a_identity[<\/span>i,<\/span> :<\/span>]<\/span>.<\/span>copy(<\/span>)<\/span>
      \n a_identity[<\/span>i,<\/span> :<\/span>]<\/span> =<\/span> a_identity[<\/span>max_row,<\/span> :<\/span>]<\/span>
      \n a_identity[<\/span>max_row,<\/span> :<\/span>]<\/span> =<\/span> temp
      \n # a_identity[[i, max_row], :] = i_identity[[max_row, i], :] # \u66ff\u6362，\u4ea4\u6362i\u884c\u548cmax_row\u884c<\/span><\/p>\n

      pivot =<\/span> a_identity[<\/span>i,<\/span> i]<\/span>
      \n if<\/span> abs<\/span>(<\/span>pivot)<\/span> <<\/span> 1e-12<\/span>:<\/span>
      \n raise<\/span> ValueError(<\/span>"\u77e9\u9635\u5947\u5f02，\u4e0d\u5b58\u5728\u9006\u77e9\u9635"<\/span>)<\/span><\/p>\n

      # \u5bf9i\u884c\u8fdb\u884c\u6d88\u5143，\u5c06i\u884c\u7684\u7b2ci\u5217\u5143\u7d20\u8bbe\u4e3a1<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>a_identity.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>
      \n a_identity[<\/span>i,<\/span> j]<\/span> =<\/span> a_identity[<\/span>i,<\/span> j]<\/span> \/<\/span> pivot
      \n # a_identity[i, :] = a_identity[i,:]\/a_identity[i, i] # \u66ff\u6362，\u5bf9i\u884c\u8fdb\u884c\u6d88\u5143，\u5c06i\u884c\u7684\u7b2ci\u5217\u5143\u7d20\u8bbe\u4e3a1<\/span><\/p>\n

      # \u5bf9\u5176\u4ed6\u884c\u8fdb\u884c\u6d88\u5143，\u5c06\u5176\u4ed6\u884c\u7684\u7b2ci\u5217\u5143\u7d20\u8bbe\u4e3a0<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
      \n if<\/span> j !=<\/span> i:<\/span>
      \n factor =<\/span> a_identity[<\/span>j,<\/span> i]<\/span> # \u2705\u63d0\u524d\u53d6\u51fa\u7cfb\u6570<\/span>
      \n for<\/span> k in<\/span> range<\/span>(<\/span>a_identity.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>
      \n a_identity[<\/span>j,<\/span> k]<\/span> =<\/span> a_identity[<\/span>j,<\/span> k]<\/span> –<\/span> a_identity[<\/span>i,<\/span> k]<\/span> *<\/span> factor
      \n # a_identity[j, :] = a_identity[j,:] – a_identity[i,:] * a_identity[j, i] # \u66ff\u6362 \u5bf9\u5176\u4ed6\u884c\u8fdb\u884c\u6d88\u5143，\u5c06\u5176\u4ed6\u884c\u7684\u7b2ci\u5217\u5143\u7d20\u8bbe\u4e3a0<\/span>
      \n return<\/span> a_identity[<\/span>:<\/span>,<\/span> n:<\/span>]<\/span><\/p>\n

      def<\/span> matrix_dot<\/span>(<\/span>a,<\/span> b)<\/span>:<\/span>
      \n # \u517c\u5bb9\u4e00\u7ef4\u5411\u91cf\u8f93\u5165<\/span>
      \n if<\/span> len<\/span>(<\/span>a.<\/span>shape)<\/span> ==<\/span> 1<\/span>:<\/span>
      \n a =<\/span> a.<\/span>reshape(<\/span>1<\/span>,<\/span> –<\/span>1<\/span>)<\/span>
      \n if<\/span> len<\/span>(<\/span>b.<\/span>shape)<\/span> ==<\/span> 1<\/span>:<\/span>
      \n b =<\/span> b.<\/span>reshape(<\/span>–<\/span>1<\/span>,<\/span> 1<\/span>)<\/span><\/p>\n

      ka =<\/span> a.<\/span>shape[<\/span>1<\/span>]<\/span>
      \n kb =<\/span> b.<\/span>shape[<\/span>0<\/span>]<\/span><\/p>\n

      if<\/span> ka !=<\/span> kb:<\/span>
      \n raise<\/span> ValueError(<\/span>f"\u77e9\u9635\u7ef4\u5ea6\u4e0d\u5339\u914d：a\u7684\u5217<\/span>{<\/span>ka}<\/span><\/span>，b\u7684\u884c<\/span>{<\/span>kb}<\/span><\/span>，\u65e0\u6cd5\u76f8\u4e58"<\/span><\/span>)<\/span><\/p>\n

      result =<\/span> np.<\/span>zeros(<\/span>(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>,<\/span> b.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>)<\/span>
      \n for<\/span> i in<\/span> range<\/span>(<\/span>a.<\/span>shape[<\/span>0<\/span>]<\/span>)<\/span>:<\/span>
      \n for<\/span> j in<\/span> range<\/span>(<\/span>b.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>
      \n s =<\/span> 0.0<\/span>
      \n for<\/span> k in<\/span> range<\/span>(<\/span>a.<\/span>shape[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>
      \n s +=<\/span> a[<\/span>i,<\/span> k]<\/span> *<\/span> b[<\/span>k,<\/span>j]<\/span>
      \n result[<\/span>i,<\/span>j]<\/span> =<\/span> s
      \n return<\/span> result<\/p>\n

      def<\/span> linear_quations_solve<\/span>(<\/span>a,<\/span> b)<\/span>:<\/span>
      \n # \u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/span>
      \n # \u5148\u6c42a\u7684\u9006\u77e9\u9635<\/span>
      \n # \u518d\u7528\u9006\u77e9\u9635\u4e58\u4ee5b<\/span>
      \n # \u5373：x = a^{-1} * b<\/span>
      \n # \u8bb0\u5f55\u539f\u59cbb\u662f\u5426\u4e00\u7ef4<\/span>
      \n is_1d =<\/span> len<\/span>(<\/span>b.<\/span>shape)<\/span> ==<\/span> 1<\/span>
      \n inverse_a =<\/span> matrix_inverse(<\/span>a)<\/span>
      \n res =<\/span> matrix_dot(<\/span>inverse_a,<\/span> b)<\/span>
      \n if<\/span> is_1d:<\/span>
      \n res =<\/span> res.<\/span>flatten(<\/span>)<\/span> # \u5c06\u7ed3\u679c\u8f6c\u6362\u4e3a\u4e00\u7ef4\u5411\u91cf，ravel()\u65b9\u6cd5\u4f1a\u6539\u53d8\u539f\u59cb\u6570\u7ec4\u7684\u5f62\u72b6<\/span>
      \n return<\/span> res<\/p>\n

      if<\/span> __name__ ==<\/span> "__main__"<\/span>:<\/span>
      \n a =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>,<\/span> [<\/span>4<\/span>,<\/span> 8<\/span>,<\/span> 6<\/span>]<\/span>,<\/span> [<\/span>7<\/span>,<\/span> 6<\/span>,<\/span> 9<\/span>]<\/span>]<\/span>)<\/span>
      \n print<\/span>(<\/span>a)<\/span>
      \n print<\/span>(<\/span>"a\u7684\u9006\u77e9\u9635\u4e3a："<\/span>)<\/span>
      \n print<\/span>(<\/span>np.<\/span>linalg.<\/span>inv(<\/span>a)<\/span>)<\/span>
      \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49\u9006\u77e9\u9635\u4e3a："<\/span>)<\/span>
      \n print<\/span>(<\/span>matrix_inverse(<\/span>a)<\/span>)<\/span>
      \n if<\/span> np.<\/span>allclose(<\/span>np.<\/span>linalg.<\/span>inv(<\/span>a)<\/span>,<\/span> matrix_inverse(<\/span>a)<\/span>)<\/span>:<\/span> # \u4e0d\u80fd\u76f4\u63a5\u7528==，\u56e0\u4e3a\u6d6e\u70b9\u6570\u6709\u7cbe\u5ea6\u95ee\u9898<\/span>
      \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49\u9006\u77e9\u9635\u8ba1\u7b97\u6b63\u786e"<\/span>)<\/span>
      \n else<\/span>:<\/span>
      \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49\u9006\u77e9\u9635\u8ba1\u7b97\u9519\u8bef"<\/span>)<\/span><\/p>\n

      #\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/span>
      \n b =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>
      \n x =<\/span> np.<\/span>linalg.<\/span>solve(<\/span>a,<\/span> b)<\/span>
      \n print<\/span>(<\/span>"\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u4e3a："<\/span>)<\/span>
      \n print<\/span>(<\/span>x)<\/span>
      \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u4e3a："<\/span>)<\/span>
      \n print<\/span>(<\/span>linear_quations_solve(<\/span>a,<\/span> b)<\/span>)<\/span><\/p>\n

      if<\/span> np.<\/span>allclose(<\/span>np.<\/span>linalg.<\/span>solve(<\/span>a,<\/span> b)<\/span>,<\/span> linear_quations_solve(<\/span>a,<\/span> b)<\/span>)<\/span>:<\/span>
      \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u6b63\u786e"<\/span>)<\/span>
      \n else<\/span>:<\/span>
      \n print<\/span>(<\/span>"\u81ea\u5b9a\u4e49\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u9519\u8bef"<\/span>)<\/span><\/p>\n

      #\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4<\/span>
      \n a1 =<\/span> np.<\/span>array(<\/span>[<\/span>[<\/span>1<\/span>,<\/span> 0<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> [<\/span>0<\/span>,<\/span> 1<\/span>,<\/span> 0<\/span>]<\/span>,<\/span> [<\/span>0<\/span>,<\/span> 0<\/span>,<\/span> 1<\/span>]<\/span>]<\/span>)<\/span>
      \n print<\/span>(<\/span>np.<\/span>linalg.<\/span>inv(<\/span>a1)<\/span>)<\/span>
      \n b1 =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>,<\/span> 2<\/span>,<\/span> 3<\/span>]<\/span>)<\/span>
      \n x1 =<\/span> np.<\/span>linalg.<\/span>solve(<\/span>a1,<\/span> b1)<\/span>
      \n print<\/span>(<\/span>"\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u4e3a："<\/span>)<\/span>
      \n print<\/span>(<\/span>x1)<\/span><\/p>\n

      print<\/span>(<\/span>"\u77e9\u9635\u7684\u9006\u77e9\u9635\u4e3a："<\/span>)<\/span>
      \n print<\/span>(<\/span>matrix_inverse(<\/span>a1)<\/span>)<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

      1. \u5f15\u8a00
      \n\u5728\u673a\u5668\u5b66\u4e60\u548c\u79d1\u5b66\u8ba1\u7b97\u4e2d，\u77e9\u9635\u6c42\u9006\u548c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u662f\u57fa\u7840\u800c\u91cd\u8981\u7684\u64cd\u4f5c\u3002\u867d\u7136 NumPy \u7b49\u5e93\u63d0\u4f9b\u4e86\u9ad8\u6548\u7684\u5b9e\u73b0，\u4f46\u7406\u89e3\u5176\u5e95\u5c42\u539f\u7406\u5bf9\u4e8e\u6df1\u5165\u638c\u63e1\u7ebf\u6027\u4ee3\u6570\u81f3\u5173\u91cd\u8981\u3002\u672c\u6587\u5c06\u5e26\u4f60\u4ece\u96f6\u5b9e\u73b0\u77e9\u9635\u6c42\u9006\u548c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\u7b97\u6cd5，\u901a\u8fc7\u4ee3\u7801\u5b9e\u8df5\u6df1\u5165\u7406\u89e3\u9ad8\u65af\u6d88\u5143\u6cd5\u7684\u539f\u7406\u3002
      \n2. \u77e9\u9635\u6c42\u9006\u7684\u5b9e\u73b0
      \n2.1 \u7b97\u6cd5\u539f\u7406
      \n\u77e9\u9635\u6c42\u9006\u7684\u6838\u5fc3\u601d\u60f3\u662f\u9ad8\u65af-\u7ea6\u65e6\u6d88\u5143\u6cd5\u3002\u5bf9\u4e8e\u4e00\u4e2a n\u00d7n \u7684\u65b9\u9635 A，\u6211\u4eec\u6784\u9020\u4e00\u4e2a\u589e\u5e7f\u77e9\u9635 [A|I]，\u5176\u4e2d I \u662f <\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[81,207,1578,1649],"topic":[],"class_list":["post-91416","post","type-post","status-publish","format-standard","hentry","category-server","tag-python","tag-207","tag-1578","tag-1649"],"yoast_head":"\n\u4ece\u96f6\u5b9e\u73b0\u77e9\u9635\u6c42\u9006\u4e0e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\uff1a\u6df1\u5165\u7406\u89e3\u9ad8\u65af\u6d88\u5143\u6cd5 - \u7f51\u7855\u4e92\u8054\u5e2e\u52a9\u4e2d\u5fc3<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.wsisp.com\/helps\/91416.html\" \/>\n<meta property=\"og:locale\" content=\"zh_CN\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u4ece\u96f6\u5b9e\u73b0\u77e9\u9635\u6c42\u9006\u4e0e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3\uff1a\u6df1\u5165\u7406\u89e3\u9ad8\u65af\u6d88\u5143\u6cd5 - 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