14.8 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;
     }

Here is the beginning of the output from the program,

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

This can be compared with the corresponding output from gnu octave,

     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;
     }

Here is the beginning of the output from the program,

     $ ./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
     ...

This can be compared with the corresponding output from gnu octave,

     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.