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<h3 class="section">32.1 Overview</h3>
<p><a name="index-root-finding_002c-overview-2206"></a>
One-dimensional root finding algorithms can be divided into two classes,
<dfn>root bracketing</dfn> and <dfn>root polishing</dfn>. Algorithms which proceed
by bracketing a root are guaranteed to converge. Bracketing algorithms
begin with a bounded region known to contain a root. The size of this
bounded region is reduced, iteratively, until it encloses the root to a
desired tolerance. This provides a rigorous error estimate for the
location of the root.
<p>The technique of <dfn>root polishing</dfn> attempts to improve an initial
guess to the root. These algorithms converge only if started “close
enough” to a root, and sacrifice a rigorous error bound for speed. By
approximating the behavior of a function in the vicinity of a root they
attempt to find a higher order improvement of an initial guess. When the
behavior of the function is compatible with the algorithm and a good
initial guess is available a polishing algorithm can provide rapid
convergence.
<p>In GSL both types of algorithm are available in similar frameworks. The
user provides a high-level driver for the algorithms, 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 solver 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 bracketing solvers is held in a <code>gsl_root_fsolver</code>
struct. The updating procedure uses only function evaluations (not
derivatives). The state for root polishing solvers is held in a
<code>gsl_root_fdfsolver</code> struct. The updates require both the function
and its derivative (hence the name <code>fdf</code>) to be supplied by the
user.
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