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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.internals1.minima"></a><a class="link" href="minima.html" title="Locating Function Minima: Brent's algorithm">Locating Function Minima:
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Brent's algorithm</a>
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</h3></div></div></div>
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<h5>
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<a name="math_toolkit.internals1.minima.h0"></a>
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<span class="phrase"><a name="math_toolkit.internals1.minima.synopsis"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.synopsis">synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">minima</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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</pre>
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<h5>
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<a name="math_toolkit.internals1.minima.h1"></a>
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<span class="phrase"><a name="math_toolkit.internals1.minima.description"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.description">Description</a>
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</h5>
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<p>
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These two functions locate the minima of the continuous function <span class="emphasis"><em>f</em></span>
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using Brent's algorithm. Parameters are:
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</p>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl class="variablelist">
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<dt><span class="term">f</span></dt>
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<dd><p>
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The function to minimise. The function should be smooth over the range
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[min,max], with no maxima occurring in that interval.
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</p></dd>
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<dt><span class="term">min</span></dt>
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<dd><p>
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The lower endpoint of the range in which to search for the minima.
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</p></dd>
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<dt><span class="term">max</span></dt>
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<dd><p>
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The upper endpoint of the range in which to search for the minima.
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</p></dd>
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<dt><span class="term">bits</span></dt>
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<dd><p>
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The number of bits precision to which the minima should be found. Note
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that in principle, the minima can not be located to greater accuracy
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than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)≅1e-8),
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therefore if <span class="emphasis"><em>bits</em></span> is set to a value greater than
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one half of the bits in type T, then the value will be ignored.
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</p></dd>
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<dt><span class="term">max_iter</span></dt>
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<dd><p>
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The maximum number of iterations to use in the algorithm, if not provided
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the algorithm will just keep on going until the minima is found.
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</p></dd>
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</dl>
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</div>
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<p>
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<span class="bold"><strong>Returns:</strong></span> a pair containing the value of
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the abscissa at the minima and the value of f(x) at the minima.
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</p>
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<h5>
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<a name="math_toolkit.internals1.minima.h2"></a>
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<span class="phrase"><a name="math_toolkit.internals1.minima.implementation"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.implementation">Implementation</a>
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</h5>
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<p>
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This is a reasonably faithful implementation of Brent's algorithm, refer
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to:
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</p>
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<p>
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Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood
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Cliffs, NJ: Prentice-Hall), Chapter 5.
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</p>
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<p>
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Numerical Recipes in C, The Art of Scientific Computing, Second Edition,
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William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.
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Flannery. Cambridge University Press. 1988, 1992.
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</p>
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<p>
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An algorithm with guaranteed convergence for finding a zero of a function,
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R. P. Brent, The Computer Journal, Vol 44, 1971.
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</p>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin
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Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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