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<title>Minimization Overview - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">33.1 Overview</h3>
<p><a name="index-minimization_002c-overview-2254"></a>
The minimization algorithms begin with a bounded region known to contain
a minimum. The region is described by a lower bound a and an
upper bound b, with an estimate of the location of the minimum
x.
<p class="noindent">The value of the function at x must be less than the value of the
function at the ends of the interval,
<pre class="example"> f(a) > f(x) < f(b)
</pre>
<p class="noindent">This condition guarantees that a minimum is contained somewhere within
the interval. On each iteration a new point x' is selected using
one of the available algorithms. If the new point is a better estimate
of the minimum, i.e. where f(x') < f(x), then the current
estimate of the minimum x 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 f(a) > f(x) <
f(b). 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>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,
<ul>
<li>initialize minimizer state, <var>s</var>, for algorithm <var>T</var>
<li>update <var>s</var> using the iteration <var>T</var>
<li>test <var>s</var> for convergence, and repeat iteration if necessary
</ul>
<p class="noindent">The state for the minimizers is held in a <code>gsl_min_fminimizer</code>
struct. The updating procedure uses only function evaluations (not
derivatives).
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