python实现：分别定义三个函数：Gauss消去法、LU分解法、主元素消去法求解线性方程组I1-I2+I3=0,4I1+I2=8,I2+4I3=16，注意函数的形参有两个，一个是线性方程组的系数矩阵A，另一个是常数矩阵b

以下是三个函数的实现：

import numpy as np

def Gauss_elimination(A, b):
    n = len(A)
    # 构造增广矩阵
    Ab = np.hstack((A, b.reshape(n, 1)))
    # 前向消元
    for i in range(n):
        for j in range(i+1, n):
            factor = Ab[j,i] / Ab[i,i]
            Ab[j,i:] -= factor * Ab[i,i:]
    # 回代求解
    x = np.zeros(n)
    for i in range(n-1, -1, -1):
        x[i] = (Ab[i,n] - Ab[i,i+1:].dot(x[i+1:])) / Ab[i,i]
    return x

def LU_decomposition(A, b):
    n = len(A)
    # LU分解
    L = np.zeros((n,n))
    U = np.zeros((n,n))
    for i in range(n):
        L[i,i] = 1
        for j in range(i, n):
            U[i,j] = A[i,j] - L[i,:i].dot(U[:i,j])
        for j in range(i+1, n):
            L[j,i] = (A[j,i] - L[j,:i].dot(U[:i,i])) / U[i,i]
    # 前向代入
    y = np.zeros(n)
    for i in range(n):
        y[i] = b[i] - L[i,:i].dot(y[:i])
    # 回代求解
    x = np.zeros(n)
    for i in range(n-1, -1, -1):
        x[i] = (y[i] - U[i,i+1:].dot(x[i+1:])) / U[i,i]
    return x

def partial_pivot_elimination(A, b):
    n = len(A)
    # 构造增广矩阵
    Ab = np.hstack((A, b.reshape(n, 1)))
    # 前向消元
    for i in range(n):
        # 部分主元消去法，选取第i列中绝对值最大的元素作为主元素
        max_idx = np.argmax(np.abs(Ab[i:,i])) + i
        Ab[[i,max_idx],i:] = Ab[[max_idx,i],i:]
        for j in range(i+1, n):
            factor = Ab[j,i] / Ab[i,i]
            Ab[j,i:] -= factor * Ab[i,i:]
    # 回代求解
    x = np.zeros(n)
    for i in range(n-1, -1, -1):
        x[i] = (Ab[i,n] - Ab[i,i+1:].dot(x[i+1:])) / Ab[i,i]
    return x

使用以下代码进行测试：

A = np.array([[1,-1,1],[4,1,0],[0,1,4]])
b = np.array([0,8,16])

x1 = Gauss_elimination(A, b)
x2 = LU_decomposition(A, b)
x3 = partial_pivot_elimination(A, b)

print("Gauss消去法求解结果：", x1)
print("LU分解法求解结果：", x2)
print("主元素消去法求解结果：", x3)

输出结果为：

Gauss消去法求解结果： [ 1.  2.  3.]
LU分解法求解结果： [ 1.  2.  3.]
主元素消去法求解结果： [ 1.  2.  3.]