高斯消元解线性方程组代码

高斯消元法是一种常见的求解线性方程组的方法，它通过一系列行操作将系数矩阵化简成阶梯形或简化阶梯形，进而找到未知数的解。以下是使用Python语言实现简单高斯消元法的代码示例：

def gauss_elimination(A, b):
    n = len(A)  # 系统的变量个数
    for i in range(n):
        # 如果当前列元素为0，则交换当前列与下一个非零列
        if A[i][i] == 0:
            non_zero_row_index = i + 1
            while non_zero_row_index < n and A[non_zero_row_index][i] == 0:
                non_zero_row_index += 1
            if non_zero_row_index < n:
                A[[i, non_zero_row_index]] = A[[non_zero_row_index, i]]
                b[[i, non_zero_row_index]] = b[[non_zero_row_index, i]]

        # 对当前列进行消元，将所有下标大于i的元素除以A[i][i]
        for j in range(i+1, n):
            ratio = A[j][i] / A[i][i]
            A[j] = [a - ratio * b for a in A[j]]
            b[j] -= ratio * b[i]

    # 检查是否得到单位矩阵，如果已经全为0，则说明无法解出唯一解或无解
    is_invertible = all(abs(val) > 1e-9 for row in A[n:] for val in row)
    
    # 回带求解
    x = [0] * n
    for i in reversed(range(n)):
        x[i] = (b[i] - sum(A[i][j] * x[j] for j in range(i))) / A[i][i]
        
    return is_invertible, x

# 示例用法
A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
b = [10, 11, 12]
solution, invertible = gauss_elimination(A, b)

if invertible:
    print(f"线性方程组有唯一解：{solution}")
else:
    print("线性方程组无解或有无穷多解")