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26 <div class=
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"title">
27 <a name=
"math_toolkit.factorials.sf_binomial"></a><a class=
"link" href=
"sf_binomial.html" title=
"Binomial Coefficients">Binomial Coefficients
</a>
28 </h3></div></div></div>
29 <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>
31 <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>
33 <span class=
"keyword">template
</span> <span class=
"special"><</span><span class=
"keyword">class
</span> <span class=
"identifier">T
</span><span class=
"special">></span>
34 <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>
36 <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=
"Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">></span>
37 <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=
"Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy
</a><span class=
"special">&);
</span>
39 <span class=
"special">}}
</span> <span class=
"comment">// namespaces
</span>
42 Returns the binomial coefficient:
<sub>n
</sub>C
<sub>k
</sub>.
48 The final
<a class=
"link" href=
"../../policy.html" title=
"Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy
</a> argument is optional and can
49 be used to control the behaviour of the function: how it handles errors,
50 what level of precision to use etc. Refer to the
<a class=
"link" href=
"../../policy.html" title=
"Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
51 documentation for more details
</a>.
54 May return the result of
<a class=
"link" href=
"../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error
</a>
55 if the result is too large to represent in type T.
57 <div class=
"important"><table border=
"0" summary=
"Important">
59 <td rowspan=
"2" align=
"center" valign=
"top" width=
"25"><img alt=
"[Important]" src=
"../../../../../../doc/src/images/important.png"></td>
60 <th align=
"left">Important
</th>
62 <tr><td align=
"left" valign=
"top">
64 The functions described above are templates where the template argument
65 T can not be deduced from the arguments passed to the function. Therefore
66 if you write something like:
69 <code class=
"computeroutput"><span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">binomial_coefficient
</span><span class=
"special">(
</span><span class=
"number">10</span><span class=
"special">,
</span> <span class=
"number">2</span><span class=
"special">);
</span></code>
72 You will get a compiler error, ususally indicating that there is no such
73 function to be found. Instead you need to specifiy the return type explicity
77 <code class=
"computeroutput"><span class=
"identifier">boost
</span><span class=
"special">::
</span><span class=
"identifier">math
</span><span class=
"special">::
</span><span class=
"identifier">binomial_coefficient
</span><span class=
"special"><</span><span class=
"keyword">double
</span><span class=
"special">>(
</span><span class=
"number">10</span><span class=
"special">,
</span> <span class=
"number">2</span><span class=
"special">);
</span></code>
80 So that the return type is known. Further, the template argument must be
81 a real-valued type such as
<code class=
"computeroutput"><span class=
"keyword">float
</span></code>
82 or
<code class=
"computeroutput"><span class=
"keyword">double
</span></code> and not an integer
83 type - that would overflow far too easily!
88 <a name=
"math_toolkit.factorials.sf_binomial.h0"></a>
89 <span class=
"phrase"><a name=
"math_toolkit.factorials.sf_binomial.accuracy"></a></span><a class=
"link" href=
"sf_binomial.html#math_toolkit.factorials.sf_binomial.accuracy">Accuracy
</a>
92 The accuracy will be the same as for the factorials for small arguments (i.e.
93 no more than one or two epsilon), and the
<a class=
"link" href=
"../sf_beta/beta_function.html" title=
"Beta">beta
</a>
94 function for larger arguments.
97 <a name=
"math_toolkit.factorials.sf_binomial.h1"></a>
98 <span class=
"phrase"><a name=
"math_toolkit.factorials.sf_binomial.testing"></a></span><a class=
"link" href=
"sf_binomial.html#math_toolkit.factorials.sf_binomial.testing">Testing
</a>
101 The spot tests for the binomial coefficients use data generated by functions.wolfram.com.
104 <a name=
"math_toolkit.factorials.sf_binomial.h2"></a>
105 <span class=
"phrase"><a name=
"math_toolkit.factorials.sf_binomial.implementation"></a></span><a class=
"link" href=
"sf_binomial.html#math_toolkit.factorials.sf_binomial.implementation">Implementation
</a>
108 Binomial coefficients are calculated using table lookup of factorials where
112 <sub>n
</sub>C
<sub>k
</sub> = n! / (k!(n-k)!)
115 Otherwise it is implemented in terms of the beta function using the relations:
118 <sub>n
</sub>C
<sub>k
</sub> =
1 / (k *
<a class=
"link" href=
"../sf_beta/beta_function.html" title=
"Beta">beta
</a>(k,
125 <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,
129 <table xmlns:
rev=
"http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width=
"100%"><tr>
130 <td align=
"left"></td>
131 <td align=
"right"><div class=
"copyright-footer">Copyright
© 2006-
2010,
2012-
2014 Nikhar Agrawal,
132 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
133 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R
åde, Gautam Sewani,
134 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang
<p>
135 Distributed under the Boost Software License, Version
1.0. (See accompanying
136 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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