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<title>GNU Scientific Library &ndash; Reference Manual: Minimization Overview</title>

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<a name="Minimization-Overview"></a>
<div class="header">
<p>
Next: <a href="Minimization-Caveats.html#Minimization-Caveats" accesskey="n" rel="next">Minimization Caveats</a>, Up: <a href="One-dimensional-Minimization.html#One-dimensional-Minimization" accesskey="u" rel="up">One dimensional Minimization</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Overview-1"></a>
<h3 class="section">35.1 Overview</h3>
<a name="index-minimization_002c-overview"></a>

<p>The minimization algorithms begin with a bounded region known to contain
a minimum.  The region is described by a lower bound <em>a</em> and an
upper bound <em>b</em>, with an estimate of the location of the minimum
<em>x</em>.
</p>

<p>The value of the function at <em>x</em> must be less than the value of the
function at the ends of the interval,
</p>
<div class="example">
<pre class="example">f(a) &gt; f(x) &lt; f(b)
</pre></div>

<p>This condition guarantees that a minimum is contained somewhere within
the interval.  On each iteration a new point <em>x'</em> is selected using
one of the available algorithms.  If the new point is a better estimate
of the minimum, i.e. where <em>f(x') &lt; f(x)</em>, then the current
estimate of the minimum <em>x</em> is updated.  The new point also allows
the size of the bounded interval to be reduced, by choosing the most
compact set of points which satisfies the constraint <em>f(a) &gt; f(x) &lt;
f(b)</em>.  The interval is reduced until it encloses the true minimum to a
desired tolerance.  This provides a best estimate of the location of the
minimum and a rigorous error estimate.
</p>
<p>Several bracketing algorithms are available within a single framework.
The user provides a high-level driver for the algorithm, and the
library provides the individual functions necessary for each of the
steps.  There are three main phases of the iteration.  The steps are,
</p>
<ul>
<li> initialize minimizer state, <var>s</var>, for algorithm <var>T</var>

</li><li> update <var>s</var> using the iteration <var>T</var>

</li><li> test <var>s</var> for convergence, and repeat iteration if necessary
</li></ul>

<p>The state for the minimizers is held in a <code>gsl_min_fminimizer</code>
struct.  The updating procedure uses only function evaluations (not
derivatives).
</p>
<hr>
<div class="header">
<p>
Next: <a href="Minimization-Caveats.html#Minimization-Caveats" accesskey="n" rel="next">Minimization Caveats</a>, Up: <a href="One-dimensional-Minimization.html#One-dimensional-Minimization" accesskey="u" rel="up">One dimensional Minimization</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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