python高斯消元法计算希尔伯特矩阵

高斯消元法是一种用于求解线性方程组的方法，而希尔伯特矩阵是一种特殊的矩阵形式。下面是使用Python实现高斯消元法计算希尔伯特矩阵的示例代码：

import numpy as np

def gaussian_elimination(matrix, vector):
    n = len(matrix)
    for i in range(n):
        # 找到主元素所在的行
        max_row = i
        for j in range(i+1, n):
            if abs(matrix[j][i]) > abs(matrix[max_row][i]):
                max_row = j
        # 交换当前行和主元素所在行
        matrix[[i, max_row]] = matrix[[max_row, i]]
        vector[[i, max_row]] = vector[[max_row, i]]
        # 消元过程
        for j in range(i+1, n):
            ratio = matrix[j][i] / matrix[i][i]
            matrix[j] -= ratio * matrix[i]
            vector[j] -= ratio * vector[i]
    
    # 回代求解
    solution = np.zeros(n)
    for i in range(n-1, -1, -1):
        solution[i] = (vector[i] - np.dot(matrix[i][i+1:], solution[i+1:])) / matrix[i][i]
    
    return solution

def hilbert_matrix(n):
    matrix = np.zeros((n, n))
    for i in range(n):
        for j in range(n):
            matrix[i][j] = 1 / (i + j + 1)
    return matrix

# 示例：计算3阶希尔伯特矩阵的解
n = 3
matrix = hilbert_matrix(n)
vector = np.ones(n)
solution = gaussian_elimination(matrix, vector)

print("希尔伯特矩阵:")
print(matrix)
print("线性方程组的解:")
print(solution)