LU分解 python

LU分解是一种矩阵分解的方法，用于求解线性方程组Ax=b。在LU分解中，矩阵A被分解为一个下三角矩阵L和一个上三角矩阵U，即A=LU。其中，L的对角线元素为1，U的对角线元素为L中每一行对应位置的系数。

以下是一个用Python实现LU分解的完整代码，并输出了L、U矩阵和解x的结果：

import numpy as np
from scipy import linalg

def LU_decomposition(A, b):
    n = len(A[0])
    L = np.zeros([n, n])
    U = np.zeros([n, n])
    
    for i in range(n):
        L[i][i] = 1
        
        if i == 0:
            U[0][0] = A[0][0]
        
        for j in range(1, n):
            U[0][j] = A[0][j]
            L[j][0] = A[j][0] / U[0][0]
        else:
            for j in range(i, n):
                temp = 0
                for k in range(0, i):
                    temp += L[i][k] * U[k][j]
                U[i][j] = A[i][j] - temp
                
            for j in range(i + 1, n):
                temp = 0
                for k in range(0, i ):
                    temp += L[j][k] * U[k][i]
                L[j][i] = (A[j][i] - temp) / U[i][i]
    
    Z = linalg.solve(L, b)
    x = linalg.solve(U, Z)
    
    print("L =", L)
    print("U =", U)
    print("Z =", Z)
    print("Exact solution: x =", x)

# 定义A、b的值
A = [[-3, -5, 10, 13], [2, 0, -4, 6], [8, -2, 1, -3], [5, -1, 1, 15]]
b = [-14, 8, 7, 0]

LU_decomposition(A, b)