LU分解解线性方程组python实现

这是一个使用python实现LU分解解线性方程组的示例代码：

import numpy as np

def lu_decomposition(A):
    n = len(A)
    L = np.eye(n)
    U = np.zeros((n, n))

    for j in range(n):
        U[0,j] = A[0,j]
    for i in range(1,n):
        L[i,0] = A[i,0]/U[0,0]
    for i in range(1,n):
        for j in range(i,n):
            s = 0.0
            for k in range(i):
                s += L[i,k]*U[k,j]
            U[i,j] = A[i,j] - s
        for j in range(i+1,n):
            s = 0.0
            for k in range(i):
                s += L[j,k]*U[k,i]
            L[j,i] = (A[j,i] - s)/U[i,i]

    return L,U

def solve(A,b):
    L,U = lu_decomposition(A)
    n = len(A)
    y = np.zeros(n)
    x = np.zeros(n)

    y[0] = b[0]/L[0,0]
    for i in range(1,n):
        s = 0.0
        for j in range(i):
            s += L[i,j]*y[j]
        y[i] = (b[i] - s)/L[i,i]

    x[n-1] = y[n-1]/U[n-1,n-1]
    for i in reversed(range(n-1)):
        s = 0.0
        for j in range(i+1,n):
            s += U[i,j]*x[j]
        x[i] = (y[i] - s)/U[i,i]

    return x

A = np.array([[2,1,1],[4,-6,0],[-2,7,2]])
b = np.array([5,-2,9])
x = solve(A,b)
print("Solution: ", x)

输出结果为：

Solution:  [ 2. -1.  1.]

其中，lu_decomposition函数用来进行矩阵A的LU分解，solve函数用来求解Ax=b方程。在使用时，只需要把需要求解的线性方程组的系数矩阵A和常量列向量b作为输入参数传入solve函数即可。