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<title>Double Factorial</title>
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<div class="section math_toolkit_special_factorials_sf_double_factorial">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.special.factorials.sf_double_factorial"></a><a class="link" href="sf_double_factorial.html" title="Double Factorial">Double
Factorial</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">factorials</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">double_factorial</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</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">double_factorial</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</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 <code class="literal">i!!</code>.
</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>
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. The implementation is
designed to be optimised for small <span class="emphasis"><em>i</em></span> where table lookup
of i! is possible.
</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">double_factorial</span><span class="special">(</span><span class="number">2</span><span class="special">);</span></code>
</p>
<p>
You will get a (possibly perplexing) 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">double_factorial</span><span class="special"><</span><span class="keyword">double</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>
<p>
The source code <code class="computeroutput"><span class="identifier">static_assert</span></code>
and comment just after the will be:
</p>
<p>
</p>
<pre class="programlisting"><span class="identifier">BOOST_STATIC_ASSERT</span><span class="special">(!</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_integral</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value</span><span class="special">);</span>
<span class="comment">// factorial<unsigned int>(n) is not implemented</span>
<span class="comment">// because it would overflow integral type T for too small n</span>
<span class="comment">// to be useful. Use instead a floating-point type,</span>
<span class="comment">// and convert to an unsigned type if essential, for example:</span>
<span class="comment">// unsigned int nfac = static_cast<unsigned int>(factorial<double>(n));</span>
<span class="comment">// See factorial documentation for more detail.</span>
</pre>
<p>
</p>
</td></tr>
</table></div>
<h5>
<a name="math_toolkit.special.factorials.sf_double_factorial.h0"></a>
<span><a name="math_toolkit.special.factorials.sf_double_factorial.accuracy"></a></span><a class="link" href="sf_double_factorial.html#math_toolkit.special.factorials.sf_double_factorial.accuracy">Accuracy</a>
</h5>
<p>
The implementation uses a trivial adaptation of the factorial function,
so error rates should be no more than a couple of epsilon higher.
</p>
<h5>
<a name="math_toolkit.special.factorials.sf_double_factorial.h1"></a>
<span><a name="math_toolkit.special.factorials.sf_double_factorial.testing"></a></span><a class="link" href="sf_double_factorial.html#math_toolkit.special.factorials.sf_double_factorial.testing">Testing</a>
</h5>
<p>
The spot tests for the double factorial use data generated by functions.wolfram.com.
</p>
<h5>
<a name="math_toolkit.special.factorials.sf_double_factorial.h2"></a>
<span><a name="math_toolkit.special.factorials.sf_double_factorial.implementation"></a></span><a class="link" href="sf_double_factorial.html#math_toolkit.special.factorials.sf_double_factorial.implementation">Implementation</a>
</h5>
<p>
The double factorial is implemented in terms of the factorial and gamma
functions using the relations:
</p>
<p>
(2n)!! = 2<sup>n </sup> * n!
</p>
<p>
(2n+1)!! = (2n+1)! / (2<sup>n </sup> n!)
</p>
<p>
and
</p>
<p>
(2n-1)!! = Γ((2n+1)/2) * 2<sup>n </sup> / sqrt(pi)
</p>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010 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>)
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