计算机数值方法：高斯消元法解线性方程组

题目

用高斯消元法解下列线性方程组（要求按三位小数计算）

高斯消元法简介

高斯消元法(Gaussian elimination)是求解线性方阵组的一种算法，它也可用来求矩阵的秩，以及求可逆方阵的逆矩阵。它通过逐步消除未知数来将原始线性系统转化为另一个更简单的等价的系统。它的实质是通过初等行变化(Elementary row operations)，将线性方程组的增广矩阵转化为行阶梯矩阵(row echelon form).

CPP代码

/*
高斯消元
WANG_zibi 20-4-4 
*/
#include <bits/stdc++.h>
using namespace std;
typedef pair<int, int> PII;
const int N = 1e6 + 10;
double eps = 1e-5;
double a[4][4] = {
    {2, 3, 4, 0},
    {1, 1, 9, 2},
    {1, 2, -6, 1}};
void test()
{
    printf("经过变换的矩阵为：\n");
    for (int i = 0; i < 3; i++)
    {
        for (int j = 0; j < 4; j++)
            cout << a[i][j] << " ";
        puts("");
    }
}
void gauss()
{
    for (int k = 0; k < 3; k++)
        for (int i = k + 1; i < 3; i++)
        {
            double t = a[i][k] / a[k][k];
            for (int j = 0; j < 4; j++)
            {
                a[i][j] -= a[k][j] * t;
            }
        }
    test();
    for (int j = 2; j >= 0; j--)
    {
        for (int k = j + 1; k < 3; k++)
        {
            a[j][3] = a[j][3] - a[j][k] * a[k][k];
        }
        a[j][j] = a[j][3] / a[j][j];
    }
}

void solve()
{
    printf("最终得到的x1,x2,x3分别为：\n");
    for (int i = 0; i < 3; i++)
        cout << a[i][i] << " ";
}
int main()
{
    gauss();
    solve();
    return 0;
}

Python代码

class gauss:
    def __init__(self, M, nn, mm):
        self.Matrix = M
        self.n = nn
        self.m = mm

    def solve(self):
        for k in range(0, self.n):
            for i in range(k+1, self.n):
                t = self.Matrix[i][k]/self.Matrix[k][k]
                for j in range(0, self.m):
                    self.Matrix[i][j] -= self.Matrix[k][j]*t
        for j in range(self.n-1,-1,-1):
            for k in range(j+1, self.n):
                self.Matrix[j][3]=self.Matrix[j][3]-self.Matrix[j][k]*self.Matrix[k][k]
            self.Matrix[j][j] = self.Matrix[j][3]/self.Matrix[j][j]
        
        for i in range(0, self.m-1):
            print(self.Matrix[i][i], end=' ')


def main():
    Matrix = [[2, 3, 4, 0], [1, 1, 9, 2], [1, 2, -6, 1]]
    G = gauss(Matrix, 3, 4)
    G.solve()
main()