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<title>GNU Scientific Library – Reference Manual: PLAIN Monte Carlo</title>
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<a name="PLAIN-Monte-Carlo"></a>
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<p>
Next: <a href="MISER.html#MISER" accesskey="n" rel="next">MISER</a>, Previous: <a href="Monte-Carlo-Interface.html#Monte-Carlo-Interface" accesskey="p" rel="previous">Monte Carlo Interface</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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<a name="PLAIN-Monte-Carlo-1"></a>
<h3 class="section">25.2 PLAIN Monte Carlo</h3>
<a name="index-plain-Monte-Carlo"></a>
<p>The plain Monte Carlo algorithm samples points randomly from the
integration region to estimate the integral and its error. Using this
algorithm the estimate of the integral <em>E(f; N)</em> for <em>N</em>
randomly distributed points <em>x_i</em> is given by,
</p>
<div class="example">
<pre class="example">E(f; N) = = V <f> = (V / N) \sum_i^N f(x_i)
</pre></div>
<p>where <em>V</em> is the volume of the integration region. The error on
this estimate <em>\sigma(E;N)</em> is calculated from the estimated
variance of the mean,
</p>
<div class="example">
<pre class="example">\sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) - <f>)^2.
</pre></div>
<p>For large <em>N</em> this variance decreases asymptotically as
<em>\Var(f)/N</em>, where <em>\Var(f)</em> is the true variance of the
function over the integration region. The error estimate itself should
decrease as <em>\sigma(f)/\sqrt{N}</em>. The familiar law of errors
decreasing as <em>1/\sqrt{N}</em> applies—to reduce the error by a
factor of 10 requires a 100-fold increase in the number of sample
points.
</p>
<p>The functions described in this section are declared in the header file
<samp>gsl_monte_plain.h</samp>.
</p>
<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005falloc"></a>Function: <em>gsl_monte_plain_state *</em> <strong>gsl_monte_plain_alloc</strong> <em>(size_t <var>dim</var>)</em></dt>
<dd><a name="index-gsl_005fmonte_005fplain_005fstate"></a>
<p>This function allocates and initializes a workspace for Monte Carlo
integration in <var>dim</var> dimensions.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005finit"></a>Function: <em>int</em> <strong>gsl_monte_plain_init</strong> <em>(gsl_monte_plain_state* <var>s</var>)</em></dt>
<dd><p>This function initializes a previously allocated integration state.
This allows an existing workspace to be reused for different
integrations.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005fintegrate"></a>Function: <em>int</em> <strong>gsl_monte_plain_integrate</strong> <em>(gsl_monte_function * <var>f</var>, const double <var>xl</var>[], const double <var>xu</var>[], size_t <var>dim</var>, size_t <var>calls</var>, gsl_rng * <var>r</var>, gsl_monte_plain_state * <var>s</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd><p>This routines uses the plain Monte Carlo algorithm to integrate the
function <var>f</var> over the <var>dim</var>-dimensional hypercubic region
defined by the lower and upper limits in the arrays <var>xl</var> and
<var>xu</var>, each of size <var>dim</var>. The integration uses a fixed number
of function calls <var>calls</var>, and obtains random sampling points using
the random number generator <var>r</var>. A previously allocated workspace
<var>s</var> must be supplied. The result of the integration is returned in
<var>result</var>, with an estimated absolute error <var>abserr</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005ffree"></a>Function: <em>void</em> <strong>gsl_monte_plain_free</strong> <em>(gsl_monte_plain_state * <var>s</var>)</em></dt>
<dd><p>This function frees the memory associated with the integrator state
<var>s</var>.
</p></dd></dl>
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