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-<div class="titlepage"><div><div><h3 class="title">
-<a name="math_toolkit.factorials.sf_double_factorial"></a><a class="link" href="sf_double_factorial.html" title="Double Factorial">Double Factorial</a>
-</h3></div></div></div>
-<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>
-<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="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">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="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy
- documentation for more details</a>.
- </p>
-<p>
- May return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.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="keyword">static_assert</span></code>
- and comment just after the will be:
- </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>
-</td></tr>
-</table></div>
-<div class="note"><table border="0" summary="Note">
-<tr>
-<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
-<th align="left">Note</th>
-</tr>
-<tr><td align="left" valign="top"><p>
- The argument to <code class="computeroutput"><span class="identifier">double_factorial</span></code>
- is type <code class="computeroutput"><span class="keyword">unsigned</span></code> even though
- technically -1!! is defined.
- </p></td></tr>
-</table></div>
-<h5>
-<a name="math_toolkit.factorials.sf_double_factorial.h0"></a>
- <span class="phrase"><a name="math_toolkit.factorials.sf_double_factorial.accuracy"></a></span><a class="link" href="sf_double_factorial.html#math_toolkit.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.factorials.sf_double_factorial.h1"></a>
- <span class="phrase"><a name="math_toolkit.factorials.sf_double_factorial.testing"></a></span><a class="link" href="sf_double_factorial.html#math_toolkit.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.factorials.sf_double_factorial.h2"></a>
- <span class="phrase"><a name="math_toolkit.factorials.sf_double_factorial.implementation"></a></span><a class="link" href="sf_double_factorial.html#math_toolkit.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>
-</div>
-<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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-<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal,
- Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
- Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani,
- Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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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