{"id":1131044,"date":"2025-01-08T20:45:21","date_gmt":"2025-01-08T12:45:21","guid":{"rendered":""},"modified":"2025-01-08T20:45:25","modified_gmt":"2025-01-08T12:45:25","slug":"python%e5%a6%82%e4%bd%95%e5%86%99%e5%90%abxy%e7%9a%84%e6%96%b9%e7%a8%8b%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1131044.html","title":{"rendered":"python\u5982\u4f55\u5199\u542bxy\u7684\u65b9\u7a0b\u5f0f"},"content":{"rendered":"

\"python\u5982\u4f55\u5199\u542bxy\u7684\u65b9\u7a0b\u5f0f\"<\/p>\n

\u4e00\u3001Python\u5982\u4f55\u5199\u542bxy\u7684\u65b9\u7a0b\u5f0f<\/strong><\/p>\n<\/p>\n

\u5728Python\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528SymPy\u5e93\u3001NumPy\u5e93\u3001\u4ee5\u53ca\u81ea\u5b9a\u4e49\u51fd\u6570\u6765\u7f16\u5199\u548c\u6c42\u89e3\u542b\u6709x\u548cy\u7684\u65b9\u7a0b\u5f0f<\/strong>\u3002SymPy\u5e93\u662fPython\u7684\u7b26\u53f7\u8ba1\u7b97\u5e93\uff0c\u7279\u522b\u9002\u5408\u5904\u7406\u6570\u5b66\u8868\u8fbe\u5f0f\u548c\u65b9\u7a0b\u5f0f\u3002NumPy\u5e93\u5219\u662f\u7528\u4e8e\u6570\u503c\u8ba1\u7b97\u7684\u5f3a\u5927\u5de5\u5177\uff0c\u9002\u7528\u4e8e\u5904\u7406\u6570\u7ec4\u548c\u6570\u503c\u8ba1\u7b97\u3002\u5728\u4f7f\u7528SymPy\u5e93\u65f6\uff0c\u6211\u4eec\u80fd\u591f\u65b9\u4fbf\u5730\u5b9a\u4e49\u7b26\u53f7\u53d8\u91cf\u3001\u7f16\u5199\u65b9\u7a0b\u5f0f\u5e76\u6c42\u89e3<\/strong>\u3002\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u8be6\u7ec6\u4ecb\u7ecd\u5982\u4f55\u4f7f\u7528\u8fd9\u4e24\u4e2a\u5e93\uff0c\u4ee5\u53ca\u5982\u4f55\u7f16\u5199\u81ea\u5b9a\u4e49\u51fd\u6570\u6765\u5904\u7406\u542bxy\u7684\u65b9\u7a0b\u5f0f\u3002<\/p>\n<\/p>\n

\u4e8c\u3001SymPy\u5e93\u7684\u4f7f\u7528<\/h3>\n<\/p>\n

1\u3001\u5b89\u88c5SymPy\u5e93<\/h4>\n<\/p>\n

SymPy\u662f\u4e00\u4e2a\u5f00\u6e90\u7684Python\u5e93\uff0c\u4e13\u95e8\u7528\u4e8e\u7b26\u53f7\u6570\u5b66\u8ba1\u7b97\u3002\u5728\u4f7f\u7528\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u5148\u5b89\u88c5\u5b83\uff1a<\/p>\n<\/p>\n

pip install sympy<\/p>\n
<\/code><\/pre>\n<\/p>\n
2\u3001\u5b9a\u4e49\u7b26\u53f7\u53d8\u91cf<\/h4>\n<\/p>\n
\u5728SymPy\u4e2d\uff0c\u9996\u5148\u9700\u8981\u5b9a\u4e49\u7b26\u53f7\u53d8\u91cf\u3002\u5bf9\u4e8e\u542b\u6709x\u548cy\u7684\u65b9\u7a0b\u5f0f\uff0c\u6211\u4eec\u9700\u8981\u5b9a\u4e49x\u548cy\u8fd9\u4e24\u4e2a\u7b26\u53f7\u53d8\u91cf\uff1a<\/p>\n<\/p>\n
from sympy import symbols<\/p>\n
x, y = symbols('x y')<\/p>\n
<\/code><\/pre>\n<\/p>\n
3\u3001\u7f16\u5199\u65b9\u7a0b\u5f0f<\/h4>\n<\/p>\n
\u5b9a\u4e49\u4e86\u7b26\u53f7\u53d8\u91cf\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u8fd9\u4e9b\u53d8\u91cf\u6765\u7f16\u5199\u65b9\u7a0b\u5f0f\u3002\u4f8b\u5982\uff0c\u4e00\u4e2a\u7b80\u5355\u7684\u4e8c\u5143\u4e00\u6b21\u65b9\u7a0b\u53ef\u4ee5\u8fd9\u6837\u8868\u793a\uff1a<\/p>\n<\/p>\n
eq = x + 2*y - 3<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u8fd9\u4e2a\u65b9\u7a0b\u8868\u793a\u7684\u662f (x + 2y = 3)\u3002<\/p>\n<\/p>\n
4\u3001\u6c42\u89e3\u65b9\u7a0b\u5f0f<\/h4>\n<\/p>\n
SymPy\u63d0\u4f9b\u4e86solve<\/code>\u51fd\u6570\u6765\u6c42\u89e3\u65b9\u7a0b\u5f0f\u3002\u5047\u8bbe\u6211\u4eec\u8981\u6c42\u89e3\u4e0a\u8ff0\u65b9\u7a0b\u5f0f\uff1a<\/p>\n<\/p>\n
from sympy import Eq, solve<\/p>\n
\u5b9a\u4e49\u65b9\u7a0b<\/strong><\/h2>\n
equation = Eq(x + 2*y, 3)<\/p>\n
\u6c42\u89e3\u65b9\u7a0b<\/strong><\/h2>\n
solution = solve(equation, x)<\/p>\n
print(solution)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e0a\u9762\u7684\u4ee3\u7801\u4f1a\u8f93\u51fax\u7684\u503c\u3002\u6211\u4eec\u4e5f\u53ef\u4ee5\u6c42\u89e3y\u7684\u503c\uff1a<\/p>\n<\/p>\n
solution = solve(equation, y)<\/p>\n
print(solution)<\/p>\n
<\/code><\/pre>\n<\/p>\n
5\u3001\u6c42\u89e3\u591a\u5143\u65b9\u7a0b\u7ec4<\/h4>\n<\/p>\n
\u5982\u679c\u6709\u591a\u4e2a\u65b9\u7a0b\u9700\u8981\u6c42\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u7ec4\u6210\u4e00\u4e2a\u65b9\u7a0b\u7ec4\uff0c\u7136\u540e\u4f7f\u7528solve<\/code>\u51fd\u6570\u6765\u6c42\u89e3\u3002\u4f8b\u5982\uff1a<\/p>\n<\/p>\n
# \u5b9a\u4e49\u65b9\u7a0b\u7ec4<\/p>\n
equation1 = Eq(x + 2*y, 3)<\/p>\n
equation2 = Eq(3*x - y, 4)<\/p>\n
\u6c42\u89e3\u65b9\u7a0b\u7ec4<\/strong><\/h2>\n
solution = solve((equation1, equation2), (x, y))<\/p>\n
print(solution)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e09\u3001NumPy\u5e93\u7684\u4f7f\u7528<\/h3>\n<\/p>\n
1\u3001\u5b89\u88c5NumPy\u5e93<\/h4>\n<\/p>\n
NumPy\u662f\u4e00\u4e2a\u7528\u4e8e\u79d1\u5b66\u8ba1\u7b97\u7684\u5f00\u6e90Python\u5e93\u3002\u5728\u4f7f\u7528\u4e4b\u524d\uff0c\u6211\u4eec\u9700\u8981\u5148\u5b89\u88c5\u5b83\uff1a<\/p>\n<\/p>\n
pip install numpy<\/p>\n
<\/code><\/pre>\n<\/p>\n
2\u3001\u5b9a\u4e49\u65b9\u7a0b\u5f0f<\/h4>\n<\/p>\n
\u5728NumPy\u4e2d\uff0c\u6211\u4eec\u901a\u5e38\u5c06\u65b9\u7a0b\u5f0f\u8868\u793a\u4e3a\u77e9\u9635\u5f62\u5f0f\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n
[ <\/p>\n
\\begin{cases} <\/p>\n
x + 2y = 3 \\<\/p>\n
3x – y = 4 <\/p>\n
\\end{cases}<\/p>\n
]<\/p>\n<\/p>\n
\u6211\u4eec\u53ef\u4ee5\u5c06\u5b83\u4eec\u8868\u793a\u4e3a\u77e9\u9635\uff1a<\/p>\n<\/p>\n
import numpy as np<\/p>\n
A = np.array([[1, 2], [3, -1]])<\/p>\n
B = np.array([3, 4])<\/p>\n
<\/code><\/pre>\n<\/p>\n
3\u3001\u6c42\u89e3\u65b9\u7a0b\u7ec4<\/h4>\n<\/p>\n
NumPy\u63d0\u4f9b\u4e86numpy.linalg.solve<\/code>\u51fd\u6570\u6765\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<\/p>\n
solution = np.linalg.solve(A, B)<\/p>\n
print(solution)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u56db\u3001\u81ea\u5b9a\u4e49\u51fd\u6570\u5b9e\u73b0<\/h3>\n<\/p>\n
\u9664\u4e86\u4f7f\u7528\u73b0\u6210\u7684\u5e93\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u7f16\u5199\u81ea\u5b9a\u4e49\u51fd\u6570\u6765\u6c42\u89e3\u65b9\u7a0b\u5f0f\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u4f7f\u7528\u725b\u987f\u8fed\u4ee3\u6cd5\u6765\u6c42\u89e3\u975e\u7ebf\u6027\u65b9\u7a0b\uff1a<\/p>\n<\/p>\n
def newton_raphson(f, df, x0, epsilon=1e-6, max_iter=100):<\/p>\n
    """<\/p>\n
    \u4f7f\u7528\u725b\u987f\u8fed\u4ee3\u6cd5\u6c42\u89e3\u65b9\u7a0b f(x) = 0<\/p>\n
    :param f: \u65b9\u7a0b<\/p>\n
    :param df: \u65b9\u7a0b\u7684\u5bfc\u6570<\/p>\n
    :param x0: \u521d\u59cb\u731c\u6d4b\u503c<\/p>\n
    :param epsilon: \u7cbe\u5ea6<\/p>\n
    :param max_iter: \u6700\u5927\u8fed\u4ee3\u6b21\u6570<\/p>\n
    :return: \u65b9\u7a0b\u7684\u6839<\/p>\n
    """<\/p>\n
    x = x0<\/p>\n
    for _ in range(max_iter):<\/p>\n
        fx = f(x)<\/p>\n
        if abs(fx) < epsilon:<\/p>\n
            return x<\/p>\n
        dfx = df(x)<\/p>\n
        if dfx == 0:<\/p>\n
            rAI<\/a>se ValueError("\u5bfc\u6570\u4e3a\u96f6\uff0c\u725b\u987f\u8fed\u4ee3\u6cd5\u5931\u8d25")<\/p>\n
        x = x - fx \/ dfx<\/p>\n
    raise ValueError("\u8fbe\u5230\u6700\u5927\u8fed\u4ee3\u6b21\u6570\uff0c\u725b\u987f\u8fed\u4ee3\u6cd5\u5931\u8d25")<\/p>\n
\u5b9a\u4e49\u65b9\u7a0b\u548c\u5bfc\u6570<\/strong><\/h2>\n
def f(x):<\/p>\n
    return x2 - 4<\/p>\n
def df(x):<\/p>\n
    return 2*x<\/p>\n
\u4f7f\u7528\u725b\u987f\u8fed\u4ee3\u6cd5\u6c42\u89e3\u65b9\u7a0b<\/strong><\/h2>\n
root = newton_raphson(f, df, x0=1)<\/p>\n
print(root)<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u4e94\u3001\u7ed3\u5408\u4f7f\u7528\u591a\u4e2a\u5de5\u5177<\/h3>\n<\/p>\n
\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u53ef\u80fd\u9700\u8981\u7ed3\u5408\u4f7f\u7528\u591a\u4e2a\u5de5\u5177\u6765\u89e3\u51b3\u590d\u6742\u7684\u65b9\u7a0b\u5f0f\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u5148\u4f7f\u7528SymPy\u6765\u6c42\u89e3\u7b26\u53f7\u89e3\uff0c\u7136\u540e\u4f7f\u7528NumPy\u6765\u8fdb\u884c\u6570\u503c\u8ba1\u7b97\u3002\u4ee5\u4e0b\u662f\u4e00\u4e2a\u793a\u4f8b\uff1a<\/p>\n<\/p>\n
from sympy import symbols, Eq, solve<\/p>\n
import numpy as np<\/p>\n
\u5b9a\u4e49\u7b26\u53f7\u53d8\u91cf<\/strong><\/h2>\n
x, y = symbols('x y')<\/p>\n
\u5b9a\u4e49\u65b9\u7a0b\u7ec4<\/strong><\/h2>\n
equation1 = Eq(x<strong>2 + y<\/strong>2, 1)<\/p>\n
equation2 = Eq(x + y, 1)<\/p>\n
\u6c42\u89e3\u65b9\u7a0b\u7ec4\u7684\u7b26\u53f7\u89e3<\/strong><\/h2>\n
symbolic_solution = solve((equation1, equation2), (x, y))<\/p>\n
print("\u7b26\u53f7\u89e3:", symbolic_solution)<\/p>\n
\u5c06\u7b26\u53f7\u89e3\u8f6c\u6362\u4e3a\u6570\u503c\u89e3<\/strong><\/h2>\n
numeric_solution = [(float(sol[0]), float(sol[1])) for sol in symbolic_solution]<\/p>\n
print("\u6570\u503c\u89e3:", numeric_solution)<\/p>\n
\u4f7f\u7528NumPy\u9a8c\u8bc1\u6570\u503c\u89e3<\/strong><\/h2>\n
for sol in numeric_solution:<\/p>\n
    x_val, y_val = sol<\/p>\n
    assert np.isclose(x_val<strong>2 + y_val<\/strong>2, 1)<\/p>\n
    assert np.isclose(x_val + y_val, 1)<\/p>\n
print("\u6570\u503c\u89e3\u9a8c\u8bc1\u901a\u8fc7")<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u516d\u3001\u7ed3\u8bba<\/h3>\n<\/p>\n
\u901a\u8fc7\u672c\u6587\u7684\u4ecb\u7ecd\uff0c\u6211\u4eec\u8be6\u7ec6\u8bb2\u89e3\u4e86\u5982\u4f55\u5728Python\u4e2d\u7f16\u5199\u548c\u6c42\u89e3\u542b\u6709x\u548cy\u7684\u65b9\u7a0b\u5f0f<\/strong>\u3002\u6211\u4eec\u9996\u5148\u4ecb\u7ecd\u4e86SymPy\u5e93\uff0c\u8fd9\u4e2a\u5e93\u63d0\u4f9b\u4e86\u5f3a\u5927\u7684\u7b26\u53f7\u8ba1\u7b97\u529f\u80fd\uff0c\u9002\u7528\u4e8e\u5904\u7406\u5404\u79cd\u6570\u5b66\u8868\u8fbe\u5f0f\u548c\u65b9\u7a0b\u3002\u63a5\u7740\uff0c\u6211\u4eec\u4ecb\u7ecd\u4e86NumPy\u5e93\uff0c\u8fd9\u4e2a\u5e93\u9002\u5408\u7528\u4e8e\u6570\u503c\u8ba1\u7b97\u548c\u7ebf\u6027\u4ee3\u6570\u3002\u6700\u540e\uff0c\u6211\u4eec\u8fd8\u5c55\u793a\u4e86\u5982\u4f55\u7f16\u5199\u81ea\u5b9a\u4e49\u51fd\u6570\u6765\u6c42\u89e3\u975e\u7ebf\u6027\u65b9\u7a0b\uff0c\u4ee5\u53ca\u5982\u4f55\u7ed3\u5408\u4f7f\u7528\u591a\u4e2a\u5de5\u5177\u6765\u89e3\u51b3\u590d\u6742\u95ee\u9898\u3002\u901a\u8fc7\u8fd9\u4e9b\u65b9\u6cd5\uff0c\u4f60\u53ef\u4ee5\u5728Python\u4e2d\u9ad8\u6548\u5730\u5904\u7406\u5404\u79cd\u542b\u6709x\u548cy\u7684\u65b9\u7a0b\u5f0f\u3002<\/p>\n<\/p>\n
\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n
 \u5982\u4f55\u5728Python\u4e2d\u8868\u793a\u548c\u89e3\u51b3\u5305\u542bxy\u7684\u65b9\u7a0b\u5f0f\uff1f<\/strong>
\u5728Python\u4e2d\uff0c\u53ef\u4ee5\u4f7f\u7528\u7b26\u53f7\u8ba1\u7b97\u5e93\u5982SymPy\u6765\u8868\u793a\u548c\u89e3\u51b3\u5305\u542bxy\u7684\u65b9\u7a0b\u5f0f\u3002\u9996\u5148\uff0c\u60a8\u9700\u8981\u5b89\u88c5SymPy\u5e93\uff0c\u53ef\u4ee5\u901a\u8fc7\u547d\u4ee4pip install sympy<\/code>\u6765\u5b8c\u6210\u3002\u7136\u540e\uff0c\u60a8\u53ef\u4ee5\u4f7f\u7528\u4ee5\u4e0b\u4ee3\u7801\u521b\u5efa\u5e76\u6c42\u89e3\u65b9\u7a0b\u5f0f\uff0c\u4f8b\u5982\uff0cy = 2x + 3<\/code>\uff1a<\/p>\n
from sympy import symbols, Eq, solve\n\nx, y = symbols('x y')\nequation = Eq(y, 2*x + 3)\nsolution = solve(equation, y)\nprint(solution)\n<\/code><\/pre>\n
\u8fd9\u5c06\u8f93\u51fay\u7684\u8868\u8fbe\u5f0f\u3002<\/p>\n
\u5728Python\u4e2d\u5982\u4f55\u7ed8\u5236\u5305\u542bxy\u7684\u65b9\u7a0b\u5f0f\u56fe\u5f62\uff1f<\/strong>
\u60a8\u53ef\u4ee5\u4f7f\u7528Matplotlib\u5e93\u6765\u7ed8\u5236\u5305\u542bxy\u7684\u65b9\u7a0b\u5f0f\u56fe\u5f62\u3002\u5b89\u88c5Matplotlib\u540e\uff0c\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u4ee3\u7801\u793a\u4f8b\u7ed8\u5236\u65b9\u7a0by = 2x + 3<\/code>\u7684\u56fe\u5f62\uff1a<\/p>\n
import numpy as np\nimport matplotlib.pyplot as plt\n\nx = np.linspace(-10, 10, 400)\ny = 2*x + 3\n\nplt.plot(x, y, label='y = 2x + 3')\nplt.title('Graph of the equation y = 2x + 3')\nplt.xlabel('x-axis')\nplt.ylabel('y-axis')\nplt.axhline(0, color='black',linewidth=0.5, ls='--')\nplt.axvline(0, color='black',linewidth=0.5, ls='--')\nplt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)\nplt.legend()\nplt.show()\n<\/code><\/pre>\n
\u6b64\u4ee3\u7801\u5c06\u751f\u6210\u65b9\u7a0b\u7684\u56fe\u5f62\uff0c\u5e2e\u52a9\u66f4\u76f4\u89c2\u5730\u7406\u89e3\u65b9\u7a0b\u7684\u884c\u4e3a\u3002<\/p>\n
Python\u4e2d\u6709\u54ea\u4e9b\u5e93\u53ef\u4ee5\u7528\u4e8e\u5904\u7406\u590d\u6742\u7684xy\u65b9\u7a0b\u5f0f\uff1f<\/strong>
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