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<a name="Root-Finding-Examples"></a>
<div class="header">
<p>
Next: <a href="Root-Finding-References-and-Further-Reading.html#Root-Finding-References-and-Further-Reading" accesskey="n" rel="next">Root Finding References and Further Reading</a>, Previous: <a href="Root-Finding-Algorithms-using-Derivatives.html#Root-Finding-Algorithms-using-Derivatives" accesskey="p" rel="previous">Root Finding Algorithms using Derivatives</a>, Up: <a href="One-dimensional-Root_002dFinding.html#One-dimensional-Root_002dFinding" accesskey="u" rel="up">One dimensional Root-Finding</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Examples-24"></a>
<h3 class="section">34.10 Examples</h3>

<p>For any root finding algorithm we need to prepare the function to be
solved.  For this example we will use the general quadratic equation
described earlier.  We first need a header file (<samp>demo_fn.h</samp>) to
define the function parameters,
</p>
<div class="example">
<pre class="verbatim">struct quadratic_params
  {
    double a, b, c;
  };

double quadratic (double x, void *params);
double quadratic_deriv (double x, void *params);
void quadratic_fdf (double x, void *params, 
                    double *y, double *dy);
</pre></div>

<p>We place the function definitions in a separate file (<samp>demo_fn.c</samp>),
</p>
<div class="example">
<pre class="verbatim">double
quadratic (double x, void *params)
{
  struct quadratic_params *p 
    = (struct quadratic_params *) params;

  double a = p-&gt;a;
  double b = p-&gt;b;
  double c = p-&gt;c;

  return (a * x + b) * x + c;
}

double
quadratic_deriv (double x, void *params)
{
  struct quadratic_params *p 
    = (struct quadratic_params *) params;

  double a = p-&gt;a;
  double b = p-&gt;b;

  return 2.0 * a * x + b;
}

void
quadratic_fdf (double x, void *params, 
               double *y, double *dy)
{
  struct quadratic_params *p 
    = (struct quadratic_params *) params;

  double a = p-&gt;a;
  double b = p-&gt;b;
  double c = p-&gt;c;

  *y = (a * x + b) * x + c;
  *dy = 2.0 * a * x + b;
}
</pre></div>

<p>The first program uses the function solver <code>gsl_root_fsolver_brent</code>
for Brent&rsquo;s method and the general quadratic defined above to solve the
following equation,
</p>
<div class="example">
<pre class="example">x^2 - 5 = 0
</pre></div>

<p>with solution <em>x = \sqrt 5 = 2.236068...</em>
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_errno.h&gt;
#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_roots.h&gt;

#include &quot;demo_fn.h&quot;
#include &quot;demo_fn.c&quot;

int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_root_fsolver_type *T;
  gsl_root_fsolver *s;
  double r = 0, r_expected = sqrt (5.0);
  double x_lo = 0.0, x_hi = 5.0;
  gsl_function F;
  struct quadratic_params params = {1.0, 0.0, -5.0};

  F.function = &amp;quadratic;
  F.params = &amp;params;

  T = gsl_root_fsolver_brent;
  s = gsl_root_fsolver_alloc (T);
  gsl_root_fsolver_set (s, &amp;F, x_lo, x_hi);

  printf (&quot;using %s method\n&quot;, 
          gsl_root_fsolver_name (s));

  printf (&quot;%5s [%9s, %9s] %9s %10s %9s\n&quot;,
          &quot;iter&quot;, &quot;lower&quot;, &quot;upper&quot;, &quot;root&quot;, 
          &quot;err&quot;, &quot;err(est)&quot;);

  do
    {
      iter++;
      status = gsl_root_fsolver_iterate (s);
      r = gsl_root_fsolver_root (s);
      x_lo = gsl_root_fsolver_x_lower (s);
      x_hi = gsl_root_fsolver_x_upper (s);
      status = gsl_root_test_interval (x_lo, x_hi,
                                       0, 0.001);

      if (status == GSL_SUCCESS)
        printf (&quot;Converged:\n&quot;);

      printf (&quot;%5d [%.7f, %.7f] %.7f %+.7f %.7f\n&quot;,
              iter, x_lo, x_hi,
              r, r - r_expected, 
              x_hi - x_lo);
    }
  while (status == GSL_CONTINUE &amp;&amp; iter &lt; max_iter);

  gsl_root_fsolver_free (s);

  return status;
}
</pre></div>

<p>Here are the results of the iterations,
</p>
<div class="smallexample">
<pre class="smallexample">$ ./a.out 
using brent method
 iter [    lower,     upper]      root        err  err(est)
    1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
    2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
    3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
    4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
    5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
Converged:                            
    6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666
</pre></div>

<p>If the program is modified to use the bisection solver instead of
Brent&rsquo;s method, by changing <code>gsl_root_fsolver_brent</code> to
<code>gsl_root_fsolver_bisection</code> the slower convergence of the
Bisection method can be observed,
</p>
<div class="smallexample">
<pre class="smallexample">$ ./a.out 
using bisection method
 iter [    lower,     upper]      root        err  err(est)
    1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
    2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
    3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
    4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
    5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
    6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
    7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
    8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
    9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
   10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
   11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
Converged:                            
   12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207
</pre></div>

<p>The next program solves the same function using a derivative solver
instead.
</p>
<div class="example">
<pre class="verbatim">#include &lt;stdio.h&gt;
#include &lt;gsl/gsl_errno.h&gt;
#include &lt;gsl/gsl_math.h&gt;
#include &lt;gsl/gsl_roots.h&gt;

#include &quot;demo_fn.h&quot;
#include &quot;demo_fn.c&quot;

int
main (void)
{
  int status;
  int iter = 0, max_iter = 100;
  const gsl_root_fdfsolver_type *T;
  gsl_root_fdfsolver *s;
  double x0, x = 5.0, r_expected = sqrt (5.0);
  gsl_function_fdf FDF;
  struct quadratic_params params = {1.0, 0.0, -5.0};

  FDF.f = &amp;quadratic;
  FDF.df = &amp;quadratic_deriv;
  FDF.fdf = &amp;quadratic_fdf;
  FDF.params = &amp;params;

  T = gsl_root_fdfsolver_newton;
  s = gsl_root_fdfsolver_alloc (T);
  gsl_root_fdfsolver_set (s, &amp;FDF, x);

  printf (&quot;using %s method\n&quot;, 
          gsl_root_fdfsolver_name (s));

  printf (&quot;%-5s %10s %10s %10s\n&quot;,
          &quot;iter&quot;, &quot;root&quot;, &quot;err&quot;, &quot;err(est)&quot;);
  do
    {
      iter++;
      status = gsl_root_fdfsolver_iterate (s);
      x0 = x;
      x = gsl_root_fdfsolver_root (s);
      status = gsl_root_test_delta (x, x0, 0, 1e-3);

      if (status == GSL_SUCCESS)
        printf (&quot;Converged:\n&quot;);

      printf (&quot;%5d %10.7f %+10.7f %10.7f\n&quot;,
              iter, x, x - r_expected, x - x0);
    }
  while (status == GSL_CONTINUE &amp;&amp; iter &lt; max_iter);

  gsl_root_fdfsolver_free (s);
  return status;
}
</pre></div>

<p>Here are the results for Newton&rsquo;s method,
</p>
<div class="example">
<pre class="example">$ ./a.out 
using newton method
iter        root        err   err(est)
    1  3.0000000 +0.7639320 -2.0000000
    2  2.3333333 +0.0972654 -0.6666667
    3  2.2380952 +0.0020273 -0.0952381
Converged:      
    4  2.2360689 +0.0000009 -0.0020263
</pre></div>

<p>Note that the error can be estimated more accurately by taking the
difference between the current iterate and next iterate rather than the
previous iterate.  The other derivative solvers can be investigated by
changing <code>gsl_root_fdfsolver_newton</code> to
<code>gsl_root_fdfsolver_secant</code> or
<code>gsl_root_fdfsolver_steffenson</code>.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Root-Finding-References-and-Further-Reading.html#Root-Finding-References-and-Further-Reading" accesskey="n" rel="next">Root Finding References and Further Reading</a>, Previous: <a href="Root-Finding-Algorithms-using-Derivatives.html#Root-Finding-Algorithms-using-Derivatives" accesskey="p" rel="previous">Root Finding Algorithms using Derivatives</a>, Up: <a href="One-dimensional-Root_002dFinding.html#One-dimensional-Root_002dFinding" accesskey="u" rel="up">One dimensional Root-Finding</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>



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