LU分解法求解方程组的代码

以下是Python代码实现LU分解法求解方程组：

import numpy as np

# LU分解函数
def LU_decomposition(A):
    n = len(A)
    L = np.eye(n)
    U = A.copy()
    for j in range(n-1):
        for i in range(j+1, n):
            L[i][j] = U[i][j] / U[j][j]
            U[i][j:] -= L[i][j] * U[j][j:]
    return L, U

# 前向替换函数
def forward_substitution(L, b):
    n = len(b)
    y = np.zeros(n)
    for i in range(n):
        y[i] = b[i] - np.dot(L[i][:i], y[:i])
    return y

# 后向替换函数
def backward_substitution(U, y):
    n = len(y)
    x = np.zeros(n)
    for i in range(n-1, -1, -1):
        x[i] = (y[i] - np.dot(U[i][i+1:], x[i+1:])) / U[i][i]
    return x

# LU分解法求解方程组
def solve_equation(A, b):
    L, U = LU_decomposition(A)
    y = forward_substitution(L, b)
    x = backward_substitution(U, y)
    return x

# 测试
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
b = np.array([1, 2, 3])
x = solve_equation(A, b)
print(x)

输出结果为：

[-0.66666667  1.33333333 -0.66666667]

表示方程组的解为 $x_1=-\frac{2}{3}$，$x_2=\frac{4}{3}$，$x_3=-\frac{2}{3}$。