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<a name="Monte-Carlo-Examples"></a>
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<p>
Next: <a href="Monte-Carlo-Integration-References-and-Further-Reading.html#Monte-Carlo-Integration-References-and-Further-Reading" accesskey="n" rel="next">Monte Carlo Integration References and Further Reading</a>, Previous: <a href="VEGAS.html#VEGAS" accesskey="p" rel="previous">VEGAS</a>, Up: <a href="Monte-Carlo-Integration.html#Monte-Carlo-Integration" accesskey="u" rel="up">Monte Carlo Integration</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Examples-17"></a>
<h3 class="section">25.5 Examples</h3>
<p>The example program below uses the Monte Carlo routines to estimate the
value of the following 3-dimensional integral from the theory of random
walks,
</p>
<div class="example">
<pre class="example">I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
\int_{-pi}^{+pi} {dk_y/(2 pi)}
\int_{-pi}^{+pi} {dk_z/(2 pi)}
1 / (1 - cos(k_x)cos(k_y)cos(k_z)).
</pre></div>
<p>The analytic value of this integral can be shown to be <em>I =
\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859...</em>. The integral gives
the mean time spent at the origin by a random walk on a body-centered
cubic lattice in three dimensions.
</p>
<p>For simplicity we will compute the integral over the region
<em>(0,0,0)</em> to <em>(\pi,\pi,\pi)</em> and multiply by 8 to obtain the
full result. The integral is slowly varying in the middle of the region
but has integrable singularities at the corners <em>(0,0,0)</em>,
<em>(0,\pi,\pi)</em>, <em>(\pi,0,\pi)</em> and <em>(\pi,\pi,0)</em>. The
Monte Carlo routines only select points which are strictly within the
integration region and so no special measures are needed to avoid these
singularities.
</p>
<div class="smallexample">
<pre class="verbatim">#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_monte.h>
#include <gsl/gsl_monte_plain.h>
#include <gsl/gsl_monte_miser.h>
#include <gsl/gsl_monte_vegas.h>
/* Computation of the integral,
I = int (dx dy dz)/(2pi)^3 1/(1-cos(x)cos(y)cos(z))
over (-pi,-pi,-pi) to (+pi, +pi, +pi). The exact answer
is Gamma(1/4)^4/(4 pi^3). This example is taken from
C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
Volume 1", Section 1.1, p21, which cites the original
paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
1800 (1977) */
/* For simplicity we compute the integral over the region
(0,0,0) -> (pi,pi,pi) and multiply by 8 */
double exact = 1.3932039296856768591842462603255;
double
g (double *k, size_t dim, void *params)
{
(void)(dim); /* avoid unused parameter warnings */
(void)(params);
double A = 1.0 / (M_PI * M_PI * M_PI);
return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
}
void
display_results (char *title, double result, double error)
{
printf ("%s ==================\n", title);
printf ("result = % .6f\n", result);
printf ("sigma = % .6f\n", error);
printf ("exact = % .6f\n", exact);
printf ("error = % .6f = %.2g sigma\n", result - exact,
fabs (result - exact) / error);
}
int
main (void)
{
double res, err;
double xl[3] = { 0, 0, 0 };
double xu[3] = { M_PI, M_PI, M_PI };
const gsl_rng_type *T;
gsl_rng *r;
gsl_monte_function G = { &g, 3, 0 };
size_t calls = 500000;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
&res, &err);
gsl_monte_plain_free (s);
display_results ("plain", res, err);
}
{
gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
&res, &err);
gsl_monte_miser_free (s);
display_results ("miser", res, err);
}
{
gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);
gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
&res, &err);
display_results ("vegas warm-up", res, err);
printf ("converging...\n");
do
{
gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
&res, &err);
printf ("result = % .6f sigma = % .6f "
"chisq/dof = %.1f\n", res, err, gsl_monte_vegas_chisq (s));
}
while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);
display_results ("vegas final", res, err);
gsl_monte_vegas_free (s);
}
gsl_rng_free (r);
return 0;
}
</pre></div>
<p>With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 1%. The estimated error <code>sigma</code> is roughly
consistent with the actual error–the computed result differs from
the true result by about 1.4 standard deviations,
</p>
<div class="example">
<pre class="example">plain ==================
result = 1.412209
sigma = 0.013436
exact = 1.393204
error = 0.019005 = 1.4 sigma
</pre></div>
<p>The <small>MISER</small> algorithm reduces the error by a factor of four, and also
correctly estimates the error,
</p>
<div class="example">
<pre class="example">miser ==================
result = 1.391322
sigma = 0.003461
exact = 1.393204
error = -0.001882 = 0.54 sigma
</pre></div>
<p>In the case of the <small>VEGAS</small> algorithm the program uses an initial
warm-up run of 10,000 function calls to prepare, or “warm up”, the grid.
This is followed by a main run with five iterations of 100,000 function
calls. The chi-squared per degree of freedom for the five iterations are
checked for consistency with 1, and the run is repeated if the results
have not converged. In this case the estimates are consistent on the
first pass.
</p>
<div class="example">
<pre class="example">vegas warm-up ==================
result = 1.392673
sigma = 0.003410
exact = 1.393204
error = -0.000531 = 0.16 sigma
converging...
result = 1.393281 sigma = 0.000362 chisq/dof = 1.5
vegas final ==================
result = 1.393281
sigma = 0.000362
exact = 1.393204
error = 0.000077 = 0.21 sigma
</pre></div>
<p>If the value of <code>chisq</code> had differed significantly from 1 it would
indicate inconsistent results, with a correspondingly underestimated
error. The final estimate from <small>VEGAS</small> (using a similar number of
function calls) is significantly more accurate than the other two
algorithms.
</p>
<hr>
<div class="header">
<p>
Next: <a href="Monte-Carlo-Integration-References-and-Further-Reading.html#Monte-Carlo-Integration-References-and-Further-Reading" accesskey="n" rel="next">Monte Carlo Integration References and Further Reading</a>, Previous: <a href="VEGAS.html#VEGAS" accesskey="p" rel="previous">VEGAS</a>, Up: <a href="Monte-Carlo-Integration.html#Monte-Carlo-Integration" accesskey="u" rel="up">Monte Carlo Integration</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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