高斯消去法（Gauss-Jordan方法）的Python实现

高斯消去法的改进形式为Gauss-Jordan Elimination Method，要求每一行的主元素所在列元素全部消去为0，除了主元素本身。区别如下：

代码实现如下：

# -*- coding: utf-8 -*-
# @Author : ZhaoKe
# @Time : 2022-09-05 23:34
from typing import List


# input a augmented matrix, output its simpler form
def GaussJordanMethod(matrix: List[List]) -> List[List]:
    PREC = 3
    m, n = len(matrix), len(matrix[0])
    # from left to right
    for j in range(n):
        # 主元素为0的话，要交换行令这一行主元素不为0
        if j >= m:
            break
        if matrix[j][j] == 0:
            for k in range(j + 1, m):
                if matrix[k][j] == 0:
                    continue
                else:
                    matrix[j], matrix[k] = matrix[k], matrix[j]
                    print(matrix)
                    break

        # from above to bottom
        # 为了实现Gauss-Jordan，pivot位置的元素应该置为1
        fac = matrix[j][j]
        for l in range(0, n):
            matrix[j][l] = matrix[j][l] / fac

        # 不再拘泥于对角线下方消除，整整一列都要消除
        for i in range(m):
            # 主元素不可以消去，直接跳过该行
            if i == j:
                continue
            # 当前行的该列元素为0的话，不必执行消去步骤，跳过即可
            if matrix[i][j] == 0:
                continue
            # replace the jth equation by a combination of itself plus a multiple of the ith equation
            coef = matrix[i][j]
            for k in range(n):
                matrix[i][k] = round(matrix[i][k] - coef * matrix[j][k], PREC)
            print(matrix[i])
    # elimination end

    print(matrix)
    # solution as follow
    for i in range(m - 1, -1, -1):
        for j in range(n - 1, -1, -1):
            if matrix[i][j] == 0:
                continue
            else:
                print("x" + str(i+1), "=", matrix[i][-1])
                break
    return matrix


if __name__ == '__main__':
    # input_m0 = [[2, 1, 1, 1], [6, 2, 1, -1], [-2, 2, 1, 7]]  # 结果正确
    # input_m1 = [[0, 1, -1, 3], [-2, 4, -1, 1], [-2, 5, -4, -2]]  # 结果正确
    input_m2 = [[2, 2, 6, 4], [2, 1, 7, 6], [-2, -6, -7, -1]]  # 结果正确
    GaussJordanMethod(input_m2)
    # 病态方程：
    # input_m3 = [[47, 28, 19], [89, 53, 36]]  # 该程序暂不适合病态方程的数值计算，另开随笔研究该问题
    # GaussJordanMethod(input_m3)

经验证该程序正确