計算：Ax-b
A: 2*2
x: 2*1
b: 2*1
so, Ax-b: 2*1

if __name__ == "__main__":
    A = np.array([[4.0, 1.0],
                  [1.0, 3.0]])
    b = np.array([[1.0], [2.0]])
    x_0 = np.array([[2.0], [1.0]])

    r_k = A * x_0 - b

    print(r_k)

錯誤！！！
修改：

if __name__ == "__main__":
    A= mat([[4.0, 1.0],
            [1.0, 3.0]]
 
)
    b = mat([[1.0], [2.0]])
    x_0 = mat([[2.0], [1.0]])

    r_k = A * x_0 - b

    print(r_k)

if __name__ == "__main__":
    A= mat([[4.0, 1.0],
            [1.0, 3.0]])
    b = mat([[1.0], [2.0]])
    x_k = mat([[2.0], [1.0]])

    p_k = -x_k  # 2行1列
    r_k = A * x_k - b # 2行1列
    alpha_k = (np.transpose(r_k) * r_k) / (np.transpose(p_k) * A * p_k) # 1
 
行1列
     -----------------------------------------------
    x_k = x_k + alpha_k * p_k #2行1列  ！！！！這裡報錯
     -----------------------------------------------
    print(x_k)

修改：

if __name__ == "__main__":
    A= mat([[4.0, 1.0],
            [1.0, 3.0]])
    b = mat([[1.0], [2.0]])
    x_k = mat([[2.0], [1.0]]
 
)

    p_k = -x_k  # 2*1
    r_k = A * x_k - b # 2*1
    alpha_k = (np.transpose(r_k) * r_k) / (np.transpose(p_k) * A * p_k) # 1*1
    -----------------------------------------------
    x_k = x_k +   p_k *alpha_k
    -----------------------------------------------
    print(x_k)