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<a name="Multimin-Overview"></a>
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<h3 class="section">35.1 Overview</h3>
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<p>The problem of multidimensional minimization requires finding a point
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x such that the scalar function,
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takes a value which is lower than at any neighboring point. For smooth
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functions the gradient g = \nabla f vanishes at the minimum. In
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general there are no bracketing methods available for the
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minimization of n-dimensional functions. The algorithms
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proceed from an initial guess using a search algorithm which attempts
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to move in a downhill direction.
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<p>Algorithms making use of the gradient of the function perform a
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one-dimensional line minimisation along this direction until the lowest
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point is found to a suitable tolerance. The search direction is then
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updated with local information from the function and its derivatives,
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and the whole process repeated until the true n-dimensional
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<p>The Nelder-Mead Simplex algorithm applies a different strategy. It
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maintains n+1 trial parameter vectors as the vertices of a
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n-dimensional simplex. In each iteration step it tries to
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improve the worst vertex by a simple geometrical transformation until
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the size of the simplex falls below a given tolerance.
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<p>Both types of algorithms use a standard framework. The user provides a
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high-level driver for the algorithms, and the library provides the
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individual functions necessary for each of the steps. There are three
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main phases of the iteration. The steps are,
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<li>initialize minimizer state, <var>s</var>, for algorithm <var>T</var>
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<li>update <var>s</var> using the iteration <var>T</var>
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<li>test <var>s</var> for convergence, and repeat iteration if necessary
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<p class="noindent">Each iteration step consists either of an improvement to the
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line-minimisation in the current direction or an update to the search
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direction itself. The state for the minimizers is held in a
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<code>gsl_multimin_fdfminimizer</code> struct or a
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<code>gsl_multimin_fminimizer</code> struct.
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