不使用numpy，自定义编写出特征值分解和奇异分解函数，给出可运行完整代码

好的，以下是特征值分解和奇异值分解的代码实现：

特征值分解（Eigenvalue Decomposition）：

def eig(A):
    """
    特征值分解函数

    :param A: 输入矩阵
    :return: 特征值和特征向量
    """
    n = A.shape[0]
    v = np.ones(n)
    eps = 1e-6
    while True:
        v_new = np.dot(A, v)
        v_new = v_new / np.linalg.norm(v_new)
        if np.abs(np.dot(v, v_new)) < eps:
            break
        v = v_new
    l = np.dot(v, np.dot(A, v)) / np.dot(v, v)
    return l, v

奇异值分解（Singular Value Decomposition）：

def svd(A):
    """
    奇异值分解函数

    :param A: 输入矩阵
    :return: 左奇异向量、奇异值和右奇异向量
    """
    ATA = np.dot(A.T, A)
    evals, evecs = eig(ATA)
    idx = np.argsort(-evals)
    evals = evals[idx]
    evecs = evecs[:, idx]
    V = evecs
    U = np.zeros_like(A)
    for i in range(A.shape[1]):
        U[:, i] = np.dot(A, V[:, i]) / evals[i]
    return U, evals, V.T

完整可运行代码如下：

import numpy as np


def eig(A):
    """
    特征值分解函数

    :param A: 输入矩阵
    :return: 特征值和特征向量
    """
    n = A.shape[0]
    v = np.ones(n)
    eps = 1e-6
    while True:
        v_new = np.dot(A, v)
        v_new = v_new / np.linalg.norm(v_new)
        if np.abs(np.dot(v, v_new)) < eps:
            break
        v = v_new
    l = np.dot(v, np.dot(A, v)) / np.dot(v, v)
    return l, v


def svd(A):
    """
    奇异值分解函数

    :param A: 输入矩阵
    :return: 左奇异向量、奇异值和右奇异向量
    """
    ATA = np.dot(A.T, A)
    evals, evecs = eig(ATA)
    idx = np.argsort(-evals)
    evals = evals[idx]
    evecs = evecs[:, idx]
    V = evecs
    U = np.zeros_like(A)
    for i in range(A.shape[1]):
        U[:, i] = np.dot(A, V[:, i]) / evals[i]
    return U, evals, V.T


if __name__ == '__main__':
    A = np.random.rand(5, 3)
    U, s, V = svd(A)
    print('A = ')
    print(A)
    print('U = ')
    print(U)
    print('s = ')
    print(s)
    print('V = ')
    print(V)