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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> &nbsp; [<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 &lt;stdio.h&gt;
#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_eigen.h&gt;

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 (&amp;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 &lt; 4; i++)
      {
        double eval_i 
           = gsl_vector_get (eval, i);
        gsl_vector_view evec_i 
           = gsl_matrix_column (evec, i);

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

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

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

octave&gt; 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 &lt;stdio.h&gt;
#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_eigen.h&gt;

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 (&amp;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 &lt; 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 (&quot;eigenvalue = %g + %gi\n&quot;,
                GSL_REAL(eval_i), GSL_IMAG(eval_i));
        printf (&quot;eigenvector = \n&quot;);
        for (j = 0; j &lt; 4; ++j)
          {
            gsl_complex z = 
              gsl_vector_complex_get(&amp;evec_i.vector, j);
            printf(&quot;%g + %gi\n&quot;, 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&gt; [v,d] = eig(vander([-1 -2 3 4]));
octave&gt; diag(d)
ans =

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

octave&gt; 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> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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