<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Binomial Coefficients</title> <link rel="stylesheet" href="../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../index.html" title="Math Toolkit"> <link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients"> <link rel="prev" href="sf_falling_factorial.html" title="Falling Factorial"> <link rel="next" href="../sf_beta.html" title="Beta Functions"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../libs/libraries.htm">Libraries</a></td> <td 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</h4></div></div></div> <p> </p> <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">special_functions</span><span class="special">/</span><span class="identifier">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> </pre> <p> </p> <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">T</span> <span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">);</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">></span> <span class="identifier">T</span> <span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Policies">Policy</a><span class="special">&);</span> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> Returns the binomial coefficient: <sub>n</sub>C<sub>k</sub>. </p> <p> Requires k <= n. </p> <p> </p> <p> The final <a class="link" href="../../policy.html" title="Policies">Policy</a> argument is optional and can be used to control the behaviour of the function: how it handles errors, what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Policies">policy documentation for more details</a>. </p> <p> </p> <p> May return the result of <a class="link" href="../../main_overview/error_handling.html#overflow_error">overflow_error</a> if the result is too large to represent in type T. </p> <a name="math_toolkit.special.factorials.sf_binomial.accuracy"></a><h5> <a name="id1092469"></a> <a class="link" href="sf_binomial.html#math_toolkit.special.factorials.sf_binomial.accuracy">Accuracy</a> </h5> <p> The accuracy will be the same as for the factorials for small arguments (i.e. no more than one or two epsilon), and the <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a> function for larger arguments. </p> <a name="math_toolkit.special.factorials.sf_binomial.testing"></a><h5> <a name="id1092491"></a> <a class="link" href="sf_binomial.html#math_toolkit.special.factorials.sf_binomial.testing">Testing</a> </h5> <p> The spot tests for the binomial coefficients use data generated by functions.wolfram.com. </p> <a name="math_toolkit.special.factorials.sf_binomial.implementation"></a><h5> <a name="id1092510"></a> <a class="link" href="sf_binomial.html#math_toolkit.special.factorials.sf_binomial.implementation">Implementation</a> </h5> <p> Binomial coefficients are calculated using table lookup of factorials where possible using: </p> <p> <sub>n</sub>C<sub>k</sub> = n! / (k!(n-k)!) </p> <p> Otherwise it is implemented in terms of the beta function using the relations: </p> <p> <sub>n</sub>C<sub>k</sub> = 1 / (k * <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>(k, n-k+1)) </p> <p> and </p> <p> <sub>n</sub>C<sub>k</sub> = 1 / ((n-k) * <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>(k+1, n-k)) </p> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying 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>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="sf_falling_factorial.html"><img src="../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../factorials.html"><img src="../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="../sf_beta.html"><img src="../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>