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<a name="Eigenvalue-and-Eigenvector-Examples"></a>
<div class="header">
<p>
Next: <a href="Eigenvalue-and-Eigenvector-References.html#Eigenvalue-and-Eigenvector-References" accesskey="n" rel="next">Eigenvalue and Eigenvector References</a>, Previous: <a href="Sorting-Eigenvalues-and-Eigenvectors.html#Sorting-Eigenvalues-and-Eigenvectors" accesskey="p" rel="previous">Sorting Eigenvalues and Eigenvectors</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-9"></a>
<h3 class="section">15.8 Examples</h3>
<p>The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, <em>H(i,j) = 1/(i + j + 1)</em>.
</p>
<div class="example">
<pre class="verbatim">#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;
}
</pre></div>
<p>Here is the beginning of the output from the program,
</p>
<div class="example">
<pre class="example">$ ./a.out
eigenvalue = 9.67023e-05
eigenvector =
-0.0291933
0.328712
-0.791411
0.514553
...
</pre></div>
<p>This can be compared with the corresponding output from <small>GNU OCTAVE</small>,
</p>
<div class="example">
<pre class="example">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
</pre></div>
<p>Note that the eigenvectors can differ by a change of sign, since the
sign of an eigenvector is arbitrary.
</p>
<p>The following program illustrates the use of the nonsymmetric
eigensolver, by computing the eigenvalues and eigenvectors of
the Vandermonde matrix
<em>V(x;i,j) = x_i^{n - j}</em>
with <em>x = (-1,-2,3,4)</em>.
</p>
<div class="example">
<pre class="verbatim">#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;
}
</pre></div>
<p>Here is the beginning of the output from the program,
</p>
<div class="example">
<pre class="example">$ ./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
...
</pre></div>
<p>This can be compared with the corresponding output from <small>GNU OCTAVE</small>,
</p>
<div class="example">
<pre class="example">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
</pre></div>
<p>Note that the eigenvectors corresponding to the eigenvalue
<em>5.54555 + 3.08545i</em> differ by the multiplicative constant
<em>0.9999984 + 0.0017674i</em> which is an arbitrary phase factor
of magnitude 1.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Eigenvalue-and-Eigenvector-References.html#Eigenvalue-and-Eigenvector-References" accesskey="n" rel="next">Eigenvalue and Eigenvector References</a>, Previous: <a href="Sorting-Eigenvalues-and-Eigenvectors.html#Sorting-Eigenvalues-and-Eigenvectors" accesskey="p" rel="previous">Sorting Eigenvalues and Eigenvectors</a>, Up: <a href="Eigensystems.html#Eigensystems" accesskey="u" rel="up">Eigensystems</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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