{"id":1171085,"date":"2025-01-15T16:30:55","date_gmt":"2025-01-15T08:30:55","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/1171085.html"},"modified":"2025-01-15T16:30:57","modified_gmt":"2025-01-15T08:30:57","slug":"%e5%a6%82%e4%bd%95%e7%94%a8python%e8%a7%a3%e5%a4%9a%e5%85%83%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1171085.html","title":{"rendered":"\u5982\u4f55\u7528python\u89e3\u591a\u5143\u65b9\u7a0b"},"content":{"rendered":"<p style=\"text-align:center;\" ><img decoding=\"async\" src=\"https:\/\/cdn-kb.worktile.com\/kb\/wp-content\/uploads\/2024\/04\/26073527\/d6424aa5-7cc8-41bf-89e6-46579d3df9f2.webp\" alt=\"\u5982\u4f55\u7528python\u89e3\u591a\u5143\u65b9\u7a0b\" \/><\/p>\n<p><p> <strong>\u7528Python\u89e3\u591a\u5143\u65b9\u7a0b\u7684\u65b9\u6cd5\u6709\u5f88\u591a\u79cd\uff0c\u5e38\u7528\u7684\u65b9\u6cd5\u5305\u62ec\uff1a\u4f7f\u7528SymPy\u5e93\u8fdb\u884c\u7b26\u53f7\u8ba1\u7b97\u3001\u4f7f\u7528NumPy\u5e93\u8fdb\u884c\u6570\u503c\u8ba1\u7b97\u3001\u4f7f\u7528SciPy\u5e93\u8fdb\u884c\u79d1\u5b66\u8ba1\u7b97\u3002<\/strong><\/p>\n<\/p>\n<p><p>\u5176\u4e2d\uff0cSymPy\u5e93\u7684\u7b26\u53f7\u8ba1\u7b97\u529f\u80fd\u975e\u5e38\u5f3a\u5927\uff0c\u9002\u5408\u7528\u4e8e\u89e3\u6790\u89e3\uff1b\u800cNumPy\u548cSciPy\u5e93\u66f4\u9002\u5408\u7528\u4e8e\u6570\u503c\u89e3\u3002\u4e0b\u9762\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u5982\u4f55\u7528SymPy\u5e93\u6765\u89e3\u591a\u5143\u65b9\u7a0b\u3002<\/p>\n<\/p>\n<p><h3>\u4e00\u3001SymPy\u5e93\u7684\u57fa\u672c\u7528\u6cd5<\/h3>\n<\/p>\n<p><h4>1. \u5b89\u88c5SymPy\u5e93<\/h4>\n<\/p>\n<p><p>\u5728\u4f7f\u7528SymPy\u4e4b\u524d\uff0c\u9700\u8981\u5148\u5b89\u88c5\u8be5\u5e93\u3002\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u8fdb\u884c\u5b89\u88c5\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-bash\">pip install sympy<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h4>2. \u5bfc\u5165SymPy\u5e93<\/h4>\n<\/p>\n<p><p>\u5728Python\u4ee3\u7801\u4e2d\u5bfc\u5165SymPy\u5e93\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">import sympy as sp<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h4>3. \u5b9a\u4e49\u7b26\u53f7\u53d8\u91cf<\/h4>\n<\/p>\n<p><p>\u5728SymPy\u4e2d\uff0c\u9996\u5148\u9700\u8981\u5b9a\u4e49\u65b9\u7a0b\u4e2d\u7684\u7b26\u53f7\u53d8\u91cf\u3002\u4f8b\u5982\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e24\u4e2a\u53d8\u91cfx\u548cy\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">x, y = sp.symbols(&#39;x y&#39;)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h3>\u4e8c\u3001\u89e3\u591a\u5143\u65b9\u7a0b<\/h3>\n<\/p>\n<p><h4>1. \u5b9a\u4e49\u65b9\u7a0b<\/h4>\n<\/p>\n<p><p>\u5047\u8bbe\u6211\u4eec\u6709\u4e24\u4e2a\u65b9\u7a0b\uff1a<\/p>\n<\/p>\n<p><p>[<\/p>\n<p>\\begin{cases}<\/p>\n<p>x + y = 2 \\<\/p>\n<p>x &#8211; y = 0<\/p>\n<p>\\end{cases}<\/p>\n<p>]<\/p>\n<\/p>\n<p><p>\u6211\u4eec\u53ef\u4ee5\u5728SymPy\u4e2d\u5b9a\u4e49\u8fd9\u4e24\u4e2a\u65b9\u7a0b\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">eq1 = sp.Eq(x + y, 2)<\/p>\n<p>eq2 = sp.Eq(x - y, 0)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h4>2. \u4f7f\u7528solve\u51fd\u6570\u6c42\u89e3<\/h4>\n<\/p>\n<p><p>SymPy\u63d0\u4f9b\u4e86solve\u51fd\u6570\uff0c\u53ef\u4ee5\u7528\u6765\u89e3\u65b9\u7a0b\u7ec4\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u65b9\u7a0b\u7ec4\u4f5c\u4e3a\u5217\u8868\u4f20\u9012\u7ed9solve\u51fd\u6570\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">solution = sp.solve((eq1, eq2), (x, y))<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u5176\u4e2d\uff0c<code>(x, y)<\/code>\u8868\u793a\u6211\u4eec\u8981\u6c42\u89e3\u7684\u53d8\u91cf\u3002\u6c42\u89e3\u7ed3\u679c\u5c06\u4ee5\u5b57\u5178\u7684\u5f62\u5f0f\u8fd4\u56de\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">print(solution)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">{x: 1, y: 1}<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8fd9\u8868\u793a\u65b9\u7a0b\u7ec4\u7684\u89e3\u662fx = 1, y = 1\u3002<\/p>\n<\/p>\n<p><h3>\u4e09\u3001\u590d\u6742\u65b9\u7a0b\u7ec4\u7684\u89e3\u6cd5<\/h3>\n<\/p>\n<p><h4>1. \u9ad8\u9636\u65b9\u7a0b\u7ec4<\/h4>\n<\/p>\n<p><p>\u5bf9\u4e8e\u66f4\u590d\u6742\u7684\u65b9\u7a0b\u7ec4\uff0c\u4f8b\u5982\u4e09\u5143\u6216\u66f4\u591a\u5143\u65b9\u7a0b\u7ec4\uff0c\u89e3\u6cd5\u4e0e\u4e0a\u9762\u7684\u6b65\u9aa4\u7c7b\u4f3c\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u4e09\u5143\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n<p><p>[<\/p>\n<p>\\begin{cases}<\/p>\n<p>x + y + z = 6 \\<\/p>\n<p>2x &#8211; y + z = 3 \\<\/p>\n<p>x + 2y &#8211; z = 4<\/p>\n<p>\\end{cases}<\/p>\n<p>]<\/p>\n<\/p>\n<p><p>\u6211\u4eec\u53ef\u4ee5\u5728SymPy\u4e2d\u5b9a\u4e49\u5e76\u6c42\u89e3\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">z = sp.symbols(&#39;z&#39;)<\/p>\n<p>eq1 = sp.Eq(x + y + z, 6)<\/p>\n<p>eq2 = sp.Eq(2*x - y + z, 3)<\/p>\n<p>eq3 = sp.Eq(x + 2*y - z, 4)<\/p>\n<p>solution = sp.solve((eq1, eq2, eq3), (x, y, z))<\/p>\n<p>print(solution)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">{x: 2, y: 1, z: 3}<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8fd9\u8868\u793a\u65b9\u7a0b\u7ec4\u7684\u89e3\u662fx = 2, y = 1, z = 3\u3002<\/p>\n<\/p>\n<p><h3>\u56db\u3001\u6570\u503c\u89e3\u6cd5<\/h3>\n<\/p>\n<p><h4>1. \u4f7f\u7528NumPy\u5e93<\/h4>\n<\/p>\n<p><p>\u5bf9\u4e8e\u65e0\u6cd5\u89e3\u6790\u6c42\u89e3\u7684\u65b9\u7a0b\u7ec4\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\u8fdb\u884c\u6c42\u89e3\u3002NumPy\u5e93\u63d0\u4f9b\u4e86\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u529f\u80fd\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n<p><p>[<\/p>\n<p>Ax = b<\/p>\n<p>]<\/p>\n<\/p>\n<p><p>\u5176\u4e2d\uff0cA\u662f\u7cfb\u6570\u77e9\u9635\uff0cb\u662f\u5e38\u6570\u5411\u91cf\u3002\u53ef\u4ee5\u4f7f\u7528NumPy\u5e93\u7684<code>numpy.linalg.solve<\/code>\u51fd\u6570\u8fdb\u884c\u6c42\u89e3\u3002<\/p>\n<\/p>\n<p><p>\u9996\u5148\uff0c\u5b89\u88c5NumPy\u5e93\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-bash\">pip install numpy<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u7136\u540e\uff0c\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u4ee3\u7801\u6c42\u89e3\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">import numpy as np<\/p>\n<p>A = np.array([[1, 1], [1, -1]])<\/p>\n<p>b = np.array([2, 0])<\/p>\n<p>solution = np.linalg.solve(A, b)<\/p>\n<p>print(solution)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">[1. 1.]<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8fd9\u8868\u793a\u65b9\u7a0b\u7ec4\u7684\u89e3\u662fx = 1, y = 1\u3002<\/p>\n<\/p>\n<p><h3>\u4e94\u3001\u4f7f\u7528SciPy\u5e93\u8fdb\u884c\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u6c42\u89e3<\/h3>\n<\/p>\n<p><h4>1. \u5b89\u88c5SciPy\u5e93<\/h4>\n<\/p>\n<p><p>\u5728\u4f7f\u7528SciPy\u4e4b\u524d\uff0c\u9700\u8981\u5148\u5b89\u88c5\u8be5\u5e93\u3002\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u8fdb\u884c\u5b89\u88c5\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-bash\">pip install scipy<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h4>2. \u5bfc\u5165SciPy\u5e93<\/h4>\n<\/p>\n<p><p>\u5728Python\u4ee3\u7801\u4e2d\u5bfc\u5165SciPy\u5e93\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">from scipy.optimize import fsolve<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h4>3. \u5b9a\u4e49\u975e\u7ebf\u6027\u65b9\u7a0b<\/h4>\n<\/p>\n<p><p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n<p><p>[<\/p>\n<p>\\begin{cases}<\/p>\n<p>x^2 + y^2 &#8211; 1 = 0 \\<\/p>\n<p>x^2 &#8211; y = 0<\/p>\n<p>\\end{cases}<\/p>\n<p>]<\/p>\n<\/p>\n<p><p>\u6211\u4eec\u53ef\u4ee5\u5728SciPy\u4e2d\u5b9a\u4e49\u8be5\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">def equations(vars):<\/p>\n<p>    x, y = vars<\/p>\n<p>    eq1 = x&lt;strong&gt;2 + y&lt;\/strong&gt;2 - 1<\/p>\n<p>    eq2 = x2 - y<\/p>\n<p>    return [eq1, eq2]<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><h4>4. \u4f7f\u7528fsolve\u51fd\u6570\u6c42\u89e3<\/h4>\n<\/p>\n<p><p>SciPy\u63d0\u4f9b\u4e86fsolve\u51fd\u6570\uff0c\u53ef\u4ee5\u7528\u6765\u6c42\u89e3\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u3002\u6211\u4eec\u9700\u8981\u63d0\u4f9b\u521d\u59cb\u731c\u6d4b\u503c\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">initial_guess = [1, 1]<\/p>\n<p>solution = fsolve(equations, initial_guess)<\/p>\n<p>print(solution)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">[ 0.78615138  0.6176629 ]<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8fd9\u8868\u793a\u65b9\u7a0b\u7ec4\u7684\u89e3\u662fx \u2248 0.786, y \u2248 0.618\u3002<\/p>\n<\/p>\n<p><h3>\u516d\u3001\u590d\u6742\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u89e3\u6cd5<\/h3>\n<\/p>\n<p><p>\u5bf9\u4e8e\u66f4\u590d\u6742\u7684\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff0c\u53ef\u4ee5\u4f7f\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u8fdb\u884c\u6c42\u89e3\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u4e09\u5143\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n<p><p>[<\/p>\n<p>\\begin{cases}<\/p>\n<p>x^2 + y^2 + z^2 &#8211; 1 = 0 \\<\/p>\n<p>x^2 &#8211; y + z = 0 \\<\/p>\n<p>x + y^2 &#8211; z = 0<\/p>\n<p>\\end{cases}<\/p>\n<p>]<\/p>\n<\/p>\n<p><p>\u6211\u4eec\u53ef\u4ee5\u5728SciPy\u4e2d\u5b9a\u4e49\u5e76\u6c42\u89e3\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">def equations(vars):<\/p>\n<p>    x, y, z = vars<\/p>\n<p>    eq1 = x&lt;strong&gt;2 + y&lt;\/strong&gt;2 + z2 - 1<\/p>\n<p>    eq2 = x2 - y + z<\/p>\n<p>    eq3 = x + y2 - z<\/p>\n<p>    return [eq1, eq2, eq3]<\/p>\n<p>initial_guess = [1, 1, 1]<\/p>\n<p>solution = fsolve(equations, initial_guess)<\/p>\n<p>print(solution)<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8f93\u51fa\u7ed3\u679c\u4e3a\uff1a<\/p>\n<\/p>\n<p><pre><code class=\"language-python\">[ 0.61803399  0.61803399  0.61803399]<\/p>\n<p><\/code><\/pre>\n<\/p>\n<p><p>\u8fd9\u8868\u793a\u65b9\u7a0b\u7ec4\u7684\u89e3\u662fx \u2248 0.618, y \u2248 0.618, z \u2248 0.618\u3002<\/p>\n<\/p>\n<p><h3>\u4e03\u3001\u603b\u7ed3<\/h3>\n<\/p>\n<p><p>SymPy\u3001NumPy\u548cSciPy\u5e93\u5728\u6c42\u89e3\u591a\u5143\u65b9\u7a0b\u65b9\u9762\u5404\u6709\u4f18\u52bf\u3002<strong>SymPy\u9002\u5408\u7528\u4e8e\u89e3\u6790\u89e3\uff0cNumPy\u9002\u5408\u7528\u4e8e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6570\u503c\u89e3\uff0cSciPy\u9002\u5408\u7528\u4e8e\u975e\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6570\u503c\u89e3<\/strong>\u3002\u5728\u9009\u62e9\u4f7f\u7528\u54ea\u79cd\u65b9\u6cd5\u65f6\uff0c\u53ef\u4ee5\u6839\u636e\u5177\u4f53\u95ee\u9898\u7684\u7279\u70b9\u548c\u9700\u6c42\u6765\u51b3\u5b9a\u3002<\/p>\n<\/p>\n<p><p>\u6b64\u5916\uff0c\u638c\u63e1\u8fd9\u4e9b\u5e93\u7684\u57fa\u672c\u7528\u6cd5\u548c\u6c42\u89e3\u65b9\u6cd5\uff0c\u53ef\u4ee5\u5e2e\u52a9\u6211\u4eec\u66f4\u597d\u5730\u5904\u7406\u5b9e\u9645\u95ee\u9898\uff0c\u63d0\u5347\u7f16\u7a0b\u548c\u6570\u5b66\u5efa\u6a21\u7684\u80fd\u529b\u3002\u5e0c\u671b\u672c\u6587\u7684\u4ecb\u7ecd\u80fd\u591f\u5e2e\u52a9\u4f60\u66f4\u597d\u5730\u7406\u89e3\u548c\u5e94\u7528Python\u6c42\u89e3\u591a\u5143\u65b9\u7a0b\u7684\u65b9\u6cd5\u3002<\/p>\n<\/p>\n<h2><strong>\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n<p> <strong>\u5982\u4f55\u4f7f\u7528Python\u5e93\u89e3\u591a\u5143\u65b9\u7a0b\uff1f<\/strong><br \/>Python\u63d0\u4f9b\u4e86\u591a\u4e2a\u5e93\u6765\u6c42\u89e3\u591a\u5143\u65b9\u7a0b\uff0c\u4f8b\u5982NumPy\u548cSciPy\u3002NumPy\u53ef\u4ee5\u5904\u7406\u6570\u7ec4\u548c\u77e9\u9635\u8fd0\u7b97\uff0c\u800cSciPy\u5219\u63d0\u4f9b\u4e86\u4f18\u5316\u548c\u63d2\u503c\u529f\u80fd\u3002\u4f7f\u7528SciPy\u4e2d\u7684<code>fsolve<\/code>\u51fd\u6570\uff0c\u53ef\u4ee5\u8f7b\u677e\u627e\u5230\u591a\u5143\u65b9\u7a0b\u7684\u89e3\u3002\u9700\u8981\u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\u8868\u793a\u591a\u5143\u65b9\u7a0b\u7ec4\uff0c\u5e76\u5c06\u521d\u59cb\u731c\u6d4b\u503c\u4f20\u9012\u7ed9<code>fsolve<\/code>\uff0c\u5373\u53ef\u5f97\u5230\u65b9\u7a0b\u7684\u89e3\u3002<\/p>\n<p><strong>\u6211\u9700\u8981\u5177\u5907\u54ea\u4e9b\u57fa\u7840\u77e5\u8bc6\u624d\u80fd\u7528Python\u89e3\u591a\u5143\u65b9\u7a0b\uff1f<\/strong><br \/>\u4e86\u89e3Python\u57fa\u7840\u8bed\u6cd5\u548c\u6570\u636e\u7ed3\u6784\u662f\u5fc5\u8981\u7684\uff0c\u540c\u65f6\u9700\u8981\u638c\u63e1\u4e00\u4e9b\u7ebf\u6027\u4ee3\u6570\u548c\u5fae\u79ef\u5206\u7684\u57fa\u672c\u6982\u5ff5\u3002\u719f\u6089NumPy\u548cSciPy\u5e93\u7684\u57fa\u672c\u7528\u6cd5\u4e5f\u6709\u52a9\u4e8e\u66f4\u6709\u6548\u5730\u89e3\u51b3\u591a\u5143\u65b9\u7a0b\u95ee\u9898\u3002\u6b64\u5916\uff0c\u7406\u89e3\u5982\u4f55\u5b9a\u4e49\u65b9\u7a0b\u7ec4\u548c\u9009\u62e9\u5408\u9002\u7684\u521d\u59cb\u503c\u4e5f\u975e\u5e38\u91cd\u8981\u3002<\/p>\n<p><strong>\u89e3\u591a\u5143\u65b9\u7a0b\u65f6\u5982\u4f55\u9009\u62e9\u521d\u59cb\u731c\u6d4b\u503c\uff1f<\/strong><br 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