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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.special.factorials.sf_binomial"></a><a class="link" href="sf_binomial.html" title="Binomial Coefficients"> Binomial |
| Coefficients</a> |
| </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> |
| <div class="important"><table border="0" summary="Important"> |
| <tr> |
| <td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../../../../../../../doc/src/images/important.png"></td> |
| <th align="left">Important</th> |
| </tr> |
| <tr><td align="left" valign="top"> |
| <p> |
| The functions described above are templates where the template argument |
| T can not be deduced from the arguments passed to the function. Therefore |
| if you write something like: |
| </p> |
| <p> |
| <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> |
| </p> |
| <p> |
| You will get a compiler error, ususally indicating that there is no such |
| function to be found. Instead you need to specifiy the return type explicity |
| and write: |
| </p> |
| <p> |
| <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> |
| </p> |
| <p> |
| So that the return type is known. Further, the template argument must |
| be a real-valued type such as <code class="computeroutput"><span class="keyword">float</span></code> |
| or <code class="computeroutput"><span class="keyword">double</span></code> and not an integer |
| type - that would overflow far too easily! |
| </p> |
| </td></tr> |
| </table></div> |
| <a name="math_toolkit.special.factorials.sf_binomial.accuracy"></a><h5> |
| <a name="id1091077"></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="id1091098"></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="id1091118"></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> |
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| 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>) |
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