C++ で連立一次方程式を解く(ガウス・ザイデル法)

考え方は、ヤコビ法とほぼ同じ.
ヤコビ法の場合、漸化式で書くと




だったが、2番目以降の式のの代わりに,
1番目の式で計算して得られたを使う.
同様に、3番目以降の式のの代わりに,
2番目の式で計算して得られたを使う.
まとめると、

#include <iostream>
#include <iomanip>
#include <math.h>
using namespace std;

void gauss(double a[4][4], double b[4], double c[4]);

int main()
{
    double a[4][4] = {{9,2,1,1},{2,8,-2,1},{-1,-2,7,-2},{1,-1,-2,6}}; 
    double b[4]    = {20,16,8,17};
    double c[4]    = {0,0,0,0};

    gauss(a,b,c);
    for (int i = 0;i<=3;i++)
        cout << setw(12) << fixed << setprecision(10) << c[i] << "\t";
    cout << endl;
   
    return 0;
}

void gauss(double a[4][4], double b[4], double c[4])
{
    for (int n = 0;n<=100;n++)
    {
        double s;
        double e = 0;
        for (int i = 0;i<=3;i++)
        {
            s = b[i];
            for (int j=0;  j<=(i-1);j++) s -= a[i][j] * c[j];
            for (int j=i+1;j<=3;    j++) s -= a[i][j] * c[j];
            s /= a[i][i];
            e += fabs(s - c[i]);
            c[i] = s;
            cout << setw(12) << fixed << setprecision(10) << c[i] << "\t";
        }
        cout << endl;

        if (e <= 0.0000000001) return;
    }
    cout << "収束しない" << endl;
}

2.2222222222	1.4444444444	1.8730158730	3.3280423280	
1.3233392122	1.7214138742	2.7746073738	3.8245482349	
1.1064462937	1.9389717407	2.9476408921	3.9546345385	
1.0244201209	1.9864758755	2.9866629927	3.9892302900	
1.0056838851	1.9965909906	2.9967609209	3.9974048246	
1.0014058081	1.9991631751	2.9992202582	3.9993663139	
1.0003430086	1.9997985232	2.9998103832	3.9998460468	
1.0000829471	1.9999511032	2.9999538924	3.9999626568	
1.0000201383	1.9999881064	2.9999888093	3.9999909311	
1.0000048941	1.9999971124	2.9999972830	3.9999977974	
1.0000011883	1.9999992990	2.9999993402	3.9999994652	
1.0000002885	1.9999998298	2.9999998398	3.9999998701	
1.0000000701	1.9999999587	2.9999999611	3.9999999685	
1.0000000170	1.9999999900	2.9999999906	3.9999999923	
1.0000000041	1.9999999976	2.9999999977	3.9999999981	
1.0000000010	1.9999999994	2.9999999994	3.9999999995	
1.0000000002	1.9999999999	2.9999999999	3.9999999999	
1.0000000001	2.0000000000	3.0000000000	4.0000000000	
1.0000000000	2.0000000000	3.0000000000	4.0000000000	
1.0000000000	2.0000000000	3.0000000000	4.0000000000	
1.0000000000	2.0000000000	3.0000000000	4.0000000000	

参考文献