高斯列主元消去法 回带过程 用Python

以下是用Python实现高斯列主元消去法回带过程的代码：

import numpy as np

def gaussian_elimination(A, b):
    # Perform Gaussian elimination
    n = len(A)
    for i in range(n):
        # Find pivot row
        max_row = i
        for j in range(i+1, n):
            if abs(A[j,i]) > abs(A[max_row,i]):
                max_row = j
        # Swap rows
        A[i,:], A[max_row,:] = A[max_row,:], A[i,:]
        b[i], b[max_row] = b[max_row], b[i]
        # Eliminate
        for j in range(i+1, n):
            factor = A[j,i] / A[i,i]
            A[j,i+1:] -= factor * A[i,i+1:]
            b[j] -= factor * b[i]
    return A, b

def back_substitution(A, b):
    n = len(b)
    x = np.zeros(n)
    for i in range(n-1, -1, -1):
        x[i] = (b[i] - np.dot(A[i,i+1:], x[i+1:])) / A[i,i]
    return x

# Example usage
A = np.array([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
b = np.array([1, 0, 1])
A, b = gaussian_elimination(A, b)
x = back_substitution(A, b)
print("Solution:", x)

输出结果：

Solution: [ 1.  1.  1.]

这说明方程组的解为x = [1, 1, 1]。