高斯消元(完全主元法 and 部分主元法) C++代码

部分主元法高斯消元

/*
算法步骤：

1.枚举每一列，找到绝对值最大的一行

2.将该行和第一行交换

3.将该行行首置为一

4.将下面所有行第 i 列置为零
*/

#include <iostream>
#include <cmath>

using namespace std;
const int N = 109;
const double eps = 1e-6;
double a[N][N];
int n;// 未知数个数， 方程数量
void init()
{
	// cout << "未知数的个数，方程的数量： ";
	cin >> n;
	for (int i = 0; i < n; i++)
		for (int j = 0; j < n + 1; j++)
			cin >> a[i][j];
}
bool Gauss()
{
	int l = 0, r = 0; //行 列
	// step 1 and 2
	for (l = 0, r = 0; r < n; r++)
	{
		int tt = l;
		for (int i = l; i < n; i++)
			if (fabs(a[tt][r]) < fabs(a[i][r]))
				tt = i;
		if (fabs(a[tt][r]) < eps)	continue;
		for (int i = r; i < n + 1; i++)
			swap(a[tt][i], a[l][i]);	

		// step 3
		for (int i = n; i >= r; i--)
			a[l][i] /= a[l][r];

		// step 4 将下面所有行第 r 列置为零
		for (int i = l + 1; i < n; i++)
		{
			if (fabs(a[i][r]) < eps)	continue;
			for (int j = n; j >= r; j--)
				a[i][j] -= a[l][j] * a[i][r];
		}
		l++;
	}
	

	for (int i = n - 1; i >= 0; i--)
		for (int j = i + 1; j < n; j++)
			a[i][n] -= a[j][n] * a[i][j];
	if (l == n)	return 0;
	return 1;
}
int main()
{
	init();
	if (!Gauss())
		for (int i = 0; i < n; i++)
			cout << "x" <<  i  << " = "<< a[i][n] << '\n';
	else
		cout << "无解！！！\n";
}

/*
3
1 1 1 6
0 4 -1 5
2 -2 1 1
*/

完全主元法高斯消元

#include <bits/stdc++.h>
using namespace std;
#define PII pair<int, int>
const int N = 109;
const double eps = 1e-6;
double a[N][N];
int id[N];
int n;

void init()
{
    cin >> n;
    for (int i = 0; i < n; i++) id[i] = i;
    for (int i = 0; i < n; i++)
        for (int j = 0; j < n + 1; j++)
            cin >> a[i][j];
}
PII Get_Max_Idx(int l, int r)
{
    PII idx;
    double mx = 0;
    for (int i = l; i < n; i++)
    for (int j = r; j < n; j++)
        if (fabs(a[i][j]) > mx)
            mx = fabs(a[i][j]), idx = {i, j};
    return idx;
}
bool Gauss()
{
    int l = 0, r = 0;

    for (l = 0, r = 0; r < n; r++)
    {
        PII tt = Get_Max_Idx(l, l);
        //行交换
        for (int i = r; i < n + 1; i++)
            swap(a[tt.first][i], a[l][i]);
        tt.first = l;
        //列交换
        for (int i = 0; i < n; i++)
            swap(a[i][tt.second], a[i][r]);
        swap(id[tt.second], id[r]);


        // step 3
        for (int i = n; i >= r; i--)
            a[l][i] /= a[l][r];

        // step 4 将下面所有行第 r 列置为零
        for (int i = l + 1; i < n; i++)
        {
            if (fabs(a[i][r]) < eps)    continue;
            for (int j = n; j >= r; j--)
                a[i][j] -= a[l][j] * a[i][r];
        }

        l++;
    }

    for (int i = n - 1; i >= 0; i--)
        for (int j = i + 1; j < n; j++)
            a[i][n] -= a[j][n] * a[i][j];

    if (l == n) return 0;
    return 1;
}
int main()
{
    init();
    if (!Gauss())
    {
        for (int i = 0; i < n; i++)
            cout << "x" <<  id[i]  << " = "<< a[i][n] << '\n';
    }
    else    cout << "无解！！！\n";
}