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<title>Overview of Multidimensional Root Finding - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">34.1 Overview</h3>
<p><a name="index-multidimensional-root-finding_002c-overview-2280"></a>
The problem of multidimensional root finding requires the simultaneous
solution of n equations, f_i, in n variables,
x_i,
<pre class="example"> f_i (x_1, ..., x_n) = 0 for i = 1 ... n.
</pre>
<p class="noindent">In general there are no bracketing methods available for n
dimensional systems, and no way of knowing whether any solutions
exist. All algorithms proceed from an initial guess using a variant of
the Newton iteration,
<pre class="example"> x -> x' = x - J^{-1} f(x)
</pre>
<p class="noindent">where x, f are vector quantities and J is the
Jacobian matrix <!-- {$J_{ij} = \partial f_i / \partial x_j$} -->
J_{ij} = d f_i / d x_j.
Additional strategies can be used to enlarge the region of
convergence. These include requiring a decrease in the norm |f| on
each step proposed by Newton's method, or taking steepest-descent steps in
the direction of the negative gradient of |f|.
<p>Several root-finding algorithms are available within a single framework.
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 evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
n^2 terms of the matrix becomes too expensive. For these reasons
the algorithms provided by the library are divided into two classes according
to whether the derivatives are available or not.
<p>The state for solvers with an analytic Jacobian matrix is held in a
<code>gsl_multiroot_fdfsolver</code> struct. The updating procedure requires
both the function and its derivatives to be supplied by the user.
<p>The state for solvers which do not use an analytic Jacobian matrix is
held in a <code>gsl_multiroot_fsolver</code> struct. The updating procedure
uses only function evaluations (not derivatives). The algorithms
estimate the matrix J or <!-- {$J^{-1}$} -->
J^{-1} by approximate methods.
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