GNU Scientific Library – Reference Manual: Eigenvalue and Eigenvector Examples

The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>

int
main (void)
{
  double data[] = { 1.0  , 1/2.0, 1/3.0, 1/4.0,
                    1/2.0, 1/3.0, 1/4.0, 1/5.0,
                    1/3.0, 1/4.0, 1/5.0, 1/6.0,
                    1/4.0, 1/5.0, 1/6.0, 1/7.0 };

  gsl_matrix_view m 
    = gsl_matrix_view_array (data, 4, 4);

  gsl_vector *eval = gsl_vector_alloc (4);
  gsl_matrix *evec = gsl_matrix_alloc (4, 4);

  gsl_eigen_symmv_workspace * w = 
    gsl_eigen_symmv_alloc (4);
  
  gsl_eigen_symmv (&m.matrix, eval, evec, w);

  gsl_eigen_symmv_free (w);

  gsl_eigen_symmv_sort (eval, evec, 
                        GSL_EIGEN_SORT_ABS_ASC);
  
  {
    int i;

    for (i = 0; i < 4; i++)
      {
        double eval_i 
           = gsl_vector_get (eval, i);
        gsl_vector_view evec_i 
           = gsl_matrix_column (evec, i);

        printf ("eigenvalue = %g\n", eval_i);
        printf ("eigenvector = \n");
        gsl_vector_fprintf (stdout, 
                            &evec_i.vector, "%g");
      }
  }

  gsl_vector_free (eval);
  gsl_matrix_free (evec);

  return 0;
}

$ ./a.out 
eigenvalue = 9.67023e-05
eigenvector = 
-0.0291933
0.328712
-0.791411
0.514553
...

octave> [v,d] = eig(hilb(4));
octave> diag(d)  
ans =

   9.6702e-05
   6.7383e-03
   1.6914e-01
   1.5002e+00

octave> v 
v =

   0.029193   0.179186  -0.582076   0.792608
  -0.328712  -0.741918   0.370502   0.451923
   0.791411   0.100228   0.509579   0.322416
  -0.514553   0.638283   0.514048   0.252161

Note that the eigenvectors can differ by a change of sign, since the sign of an eigenvector is arbitrary.

The following program illustrates the use of the nonsymmetric eigensolver, by computing the eigenvalues and eigenvectors of the Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).

#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>

int
main (void)
{
  double data[] = { -1.0, 1.0, -1.0, 1.0,
                    -8.0, 4.0, -2.0, 1.0,
                    27.0, 9.0, 3.0, 1.0,
                    64.0, 16.0, 4.0, 1.0 };

  gsl_matrix_view m 
    = gsl_matrix_view_array (data, 4, 4);

  gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
  gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);

  gsl_eigen_nonsymmv_workspace * w = 
    gsl_eigen_nonsymmv_alloc (4);
  
  gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);

  gsl_eigen_nonsymmv_free (w);

  gsl_eigen_nonsymmv_sort (eval, evec, 
                           GSL_EIGEN_SORT_ABS_DESC);
  
  {
    int i, j;

    for (i = 0; i < 4; i++)
      {
        gsl_complex eval_i 
           = gsl_vector_complex_get (eval, i);
        gsl_vector_complex_view evec_i 
           = gsl_matrix_complex_column (evec, i);

        printf ("eigenvalue = %g + %gi\n",
                GSL_REAL(eval_i), GSL_IMAG(eval_i));
        printf ("eigenvector = \n");
        for (j = 0; j < 4; ++j)
          {
            gsl_complex z = 
              gsl_vector_complex_get(&evec_i.vector, j);
            printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
          }
      }
  }

  gsl_vector_complex_free(eval);
  gsl_matrix_complex_free(evec);

  return 0;
}

$ ./a.out 
eigenvalue = -6.41391 + 0i
eigenvector = 
-0.0998822 + 0i
-0.111251 + 0i
0.292501 + 0i
0.944505 + 0i
eigenvalue = 5.54555 + 3.08545i
eigenvector = 
-0.043487 + -0.0076308i
0.0642377 + -0.142127i
-0.515253 + 0.0405118i
-0.840592 + -0.00148565i
...

octave> [v,d] = eig(vander([-1 -2 3 4]));
octave> diag(d)
ans =

  -6.4139 + 0.0000i
   5.5456 + 3.0854i
   5.5456 - 3.0854i
   2.3228 + 0.0000i

octave> v
v =

 Columns 1 through 3:

  -0.09988 + 0.00000i  -0.04350 - 0.00755i  -0.04350 + 0.00755i
  -0.11125 + 0.00000i   0.06399 - 0.14224i   0.06399 + 0.14224i
   0.29250 + 0.00000i  -0.51518 + 0.04142i  -0.51518 - 0.04142i
   0.94451 + 0.00000i  -0.84059 + 0.00000i  -0.84059 - 0.00000i

 Column 4:

  -0.14493 + 0.00000i
   0.35660 + 0.00000i
   0.91937 + 0.00000i
   0.08118 + 0.00000i

Note that the eigenvectors corresponding to the eigenvalue 5.54555 + 3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i which is an arbitrary phase factor of magnitude 1.

15.8 Examples