C++で固有値・固有ベクトルを求める (反復法)

n × n の正方行列 A と
n次元のベクトル x について
Ax = λx (ただし x ≠ 0) が成り立つとき
λを固有値, x を固有ベクトルという.
最初に適当なベクトルx₀から始めて x_k+1 = Ax_k を反復すると
x_k は行列 A の最大固有値に対応する固有ベクトルに収束する.
固有値はレイリー(Rayleigh)商
λ = (x_k^Tx_k+1) / (x_k^Tx_k)
により求める.
べき乗法、累乗法とも言う.

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

const int N = 4;

// ヤコビの反復法
void jacobi(double a[N][N], double b[N], double c[N]);
// １次元配列を表示
void disp_vector(double row[N]);

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

    // ヤコビの反復法
    jacobi(a,b,c);

    cout << "X" << endl;
    disp_vector(c);

    return 0;
}

// ヤコビの反復法
void jacobi(double a[N][N], double b[N], double x0[N])
{
    while (true)
    {
        double x1[N];
        bool finish = true;
        for (int i = 0; i < N; i++)
        {
            x1[i] = 0;
            for (int j = 0; j < N; j++)
                if (j != i)
                    x1[i] += a[i][j] * x0[j];

            x1[i] = (b[i] - x1[i]) / a[i][i];
            if (fabs(x1[i] - x0[i]) > 0.0000000001) finish = false;
        }
        for (int i = 0; i < N; i++)
            x0[i] = x1[i];
        if (finish) return;

        disp_vector(x0);
    }
}

// １次元配列を表示
void disp_vector(double row[N])
{
    for (int i = 0; i < N; i++)
        cout << setw(14) << fixed << setprecision(10) << row[i] << "\t";
    cout << endl;
}

z:\cpp>bcc32 -q CP1101.cpp
cp1101.cpp:

z:\cpp>CP1101
  1       1.0000000000    0.0000000000    0.0000000000    0.0000000000
  2       0.7624928517    0.6099942813    0.1524985703    0.1524985703
  3       0.6745979924    0.6589096670    0.2353248811    0.2353248811
  4       0.6517382447    0.6501601860    0.2761602732    0.2761602732
  5       0.6421032522    0.6419452154    0.2963188771    0.2963188771
  6       0.6373267026    0.6373108932    0.3063082594    0.3063082594
  7       0.6349074765    0.6349058954    0.3112772079    0.3112772079
  8       0.6336860412    0.6336858831    0.3137548428    0.3137548428
  9       0.6330719648    0.6330719490    0.3149919005    0.3149919005
 10       0.6327640473    0.6327640457    0.3156099832    0.3156099832
 11       0.6326098648    0.6326098646    0.3159189122    0.3159189122
 12       0.6325327172    0.6325327172    0.3160733485    0.3160733485
 13       0.6324941293    0.6324941293    0.3161505596    0.3161505596
 14       0.6324748319    0.6324748319    0.3161891634    0.3161891634
 15       0.6324651822    0.6324651822    0.3162084649    0.3162084649
 16       0.6324603572    0.6324603572    0.3162181155    0.3162181155
 17       0.6324579446    0.6324579446    0.3162229408    0.3162229408
 18       0.6324567383    0.6324567383    0.3162253534    0.3162253534
 19       0.6324561352    0.6324561352    0.3162265597    0.3162265597
 20       0.6324558336    0.6324558336    0.3162271629    0.3162271629
 21       0.6324556828    0.6324556828    0.3162274644    0.3162274644
 22       0.6324556074    0.6324556074    0.3162276152    0.3162276152
 23       0.6324555697    0.6324555697    0.3162276906    0.3162276906
 24       0.6324555509    0.6324555509    0.3162277283    0.3162277283
 25       0.6324555415    0.6324555415    0.3162277472    0.3162277472
 26       0.6324555367    0.6324555367    0.3162277566    0.3162277566
 27       0.6324555344    0.6324555344    0.3162277613    0.3162277613
 28       0.6324555332    0.6324555332    0.3162277637    0.3162277637
 29       0.6324555326    0.6324555326    0.3162277648    0.3162277648
 30       0.6324555323    0.6324555323    0.3162277654    0.3162277654
 31       0.6324555322    0.6324555322    0.3162277657    0.3162277657
 32       0.6324555321    0.6324555321    0.3162277659    0.3162277659
 33       0.6324555321    0.6324555321    0.3162277659    0.3162277659
 34       0.6324555321    0.6324555321    0.3162277660    0.3162277660
 35       0.6324555320    0.6324555320    0.3162277660    0.3162277660

eigenvalue
 10.0000000000
eigenvector
  0.6324555320    0.6324555320    0.3162277660    0.3162277660

参考文献