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 GNU Scientific Library Reference Manual - Third Edition (v1.12) by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. RossiPaperback (6"x9"), 592 pages, 60 figuresISBN 0954612078RRP £24.95 (\$39.95)

## 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.

 ISBN 0954612078 GNU Scientific Library Reference Manual - Third Edition (v1.12) See the print edition