不选主元Gauss消去法

1.不选主元Gauss消去法原理

设Ax=b,其中A∈R^n×n.
(1) 如果a_kk^(k)≠0(k=1,2,···,n),则可以通过高斯消去法将Ax=b约化为等价的三角形线性方程组，如下:
$$\left[\begin{array}{cccc}{a}_{11}^{(1)}& a_{12}^{(1)}& \dots & a_{1n}^{(1)}\\ & a_{22}^{(2)}& \dots & a_{2n}^{(2)}\\ & & \ddots & ⋮\\ & & & a_{nn}^{(n)}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]=\left[\begin{array}{c}{b}_1^{(1)}\\ b_2^{(2)}\\ ⋮\\ b_n^{(n)}\end{array}\right](1)$$
且计算公式为:

1. 消元计算$(k=1,2,\dots ,n-1)$$$\{\begin{array}{l}{m}_{ik}=a_{ik}^{(k)}/a_{kk}^{(k)},i=k+1,\dots ,n,\\ a_{ij}^{(k+1)}=a_{ij}^{(k)}-m_{ik}{a}_{kj}^{(k)},i,j=k+1,\dots ,n,\\ b_i^{(k+1)}=b_i^{(k)}-m_{ik}{b}_k^{(k)},i=k+1,\dots ,n.\end{array}(2)$$
2. 回代计算$$\{\begin{array}{l}{x}_n=b_n^{(n)}/a_{nn}^{(n)},\\ x_i=(b_i^{(i)}-\sum _{j=i+1}^n{a}_{ij}^{(i)}{x}_j)/a_{ii}^{(i)},i=n-1,\dots ,2,1.\end{array}(3)$$

（2）如果$A$为非奇异矩阵，则可通过高斯消去法(及交换两行的初等变换)将方程组$Ax=b$约化为方程组(1).

2.Python实现不选主元Gauss消去

# 自己原创
def lu_decomposition(coefficient_matrix: np.ndarray, right_hand_side_vector: np.ndarray):
    """
    实现方程Ax=b系数矩阵A的LU decomposition
    :param coefficient_matrix: 初始系数矩阵A
    :param right_hand_side_vector: 初始常数列向量b
    :return: 单位下三角矩阵L，上三角矩阵U，常数列向量b
    """
    # first step: evaluate the lu decomposition condition
    rows, columns = coefficient_matrix.shape
    if rows == columns:  # judge if it is a square matrix
        for k in range(rows):  # 判断各阶顺序主子式是否为0
            if det(coefficient_matrix[:k + 1, :k + 1]) == 0:
                raise Exception("cannot generate LU decomposition")
        else:
            # LU decomposition
            lower_triangular_matrix = np.eye(rows)
            for k in range(rows - 1):
                for row in range(k + 1, rows):
                    multiplier = coefficient_matrix[row, k] / coefficient_matrix[k, k]
                    coefficient_matrix[row, k:] += -multiplier * (coefficient_matrix[k, k:])
                    right_hand_side_vector[row] += -multiplier * right_hand_side_vector[k]
                    lower_triangular_matrix[row, k] = multiplier
    else:
        raise Exception("ERROR:please pass a square matrix.")

    return lower_triangular_matrix, coefficient_matrix, right_hand_side_vector
# 自己原创
def gaussian_elimination(coefficient_matrix: np.ndarray,
                         right_hand_side_vector: np.ndarray,
                         decomposition_function=lu_decomposition):
    """
    实现高斯消元法解方程Ax=b
    :param decomposition_function:分解函数
    :param coefficient_matrix: 初始系数矩阵A
    :param right_hand_side_vector: 初始列向量b
    :return: 解向量x
    """
    *_, upper_triangular_matrix, constant_column_vector = decomposition_function(coefficient_matrix,
                                                                                 right_hand_side_vector)
    rows = right_hand_side_vector.shape[0]
    if upper_triangular_matrix.shape[0] == rows:  # 判断上三角矩阵的行数是否等于列向量行数
        # 此处不指定双精度浮点数据类型，则可能会损失精度，导致结果偏差太大
        variable_column_vector = np.ones_like(right_hand_side_vector, dtype=np.float64)  # 构建一个与列向量同形状的1矩阵
        for row in range(rows - 1, -1, -1):
            prod = 0  # 声明累积变量，初始化为0
            for column in range(row + 1, rows):
                prod += variable_column_vector[column] * upper_triangular_matrix[row, column]
            # 回代运算得到解向量
            variable_column_vector[row] = (constant_column_vector[row] - prod) / upper_triangular_matrix[row, row]
        return variable_column_vector
    else:
        raise Exception("系数矩阵的行数与常数列向量行数不相等")

3.解方程

对比系统LU分解函数和不选主元高斯消去法解线性方程组
$$\left[\begin{array}{cccc}10& -7& 0& 1\\ -3& 2.099999& 6& 2\\ 5& -1& 5& -1\\ 2& 1& 0& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}8\\ 5.900001\\ 5\\ 1\end{array}\right]$$

4.测试

if __name__ == '__main__':
	square_matrix = np.array([[10, -7, 0, 1], [-3, 2.099999, 6, 2], [5, -1, 5, -1], [2, 1, 0, 2]])
    vector = np.array([8, 5.900001, 5, 1]).reshape((4, 1))
    copy_square_matrix1 = square_matrix.copy()
    copy_vector1 = vector.copy()
    copy_square_matrix2 = square_matrix.copy()
    copy_vector2 = vector.copy()
    solution = lu_solve(lu_factor(square_matrix), vector)
    print("系统函数LU分解实现的解向量x=\n{}".format(solution))
    solution_vector = gaussian_elimination(copy_square_matrix2, copy_vector2)
    print("自己写的不选主元高斯消去法解向量x=\n{}".format(solution_vector))

5.运行截图

说明：计算机底层极小的浮点数满足精度情况下认为就是0，故解向量正确