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  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.factorials.sf_double_factorial"></a><a class="link" href="sf_double_factorial.html" title="Double Factorial">Double Factorial</a>
  28 </h3></div></div></div>
  29 <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</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">&gt;</span>
  30 </pre>
  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>
  32
  33 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
  34 <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>
  35
  36 <span class="keyword">template</span> <span class="special">&lt;</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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  37 <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&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  38
  39 <span class="special">}}</span> <span class="comment">// namespaces</span>
  40 </pre>
  41 <p>
  42         Returns <code class="literal">i!!</code>.
  43       </p>
  44 <p>
  45         The final <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  46         be used to control the behaviour of the function: how it handles errors,
  47         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;14.&#160;Policies: Controlling Precision, Error Handling etc">policy
  48         documentation for more details</a>.
  49       </p>
  50 <p>
  51         May return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
  52         if the result is too large to represent in type T. The implementation is
  53         designed to be optimised for small <span class="emphasis"><em>i</em></span> where table lookup
  54         of i! is possible.
  55       </p>
  56 <div class="important"><table border="0" summary="Important">
  57 <tr>
  58 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Important]" src="../../../../../../doc/src/images/important.png"></td>
  59 <th align="left">Important</th>
  60 </tr>
  61 <tr><td align="left" valign="top">
  62 <p>
  63           The functions described above are templates where the template argument
  64           T can not be deduced from the arguments passed to the function. Therefore
  65           if you write something like:
  66         </p>
  67 <p>
  68           <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>
  69         </p>
  70 <p>
  71           You will get a (possibly perplexing) compiler error, ususally indicating
  72           that there is no such function to be found. Instead you need to specifiy
  73           the return type explicity and write:
  74         </p>
  75 <p>
  76           <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">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">2</span><span class="special">);</span></code>
  77         </p>
  78 <p>
  79           So that the return type is known. Further, the template argument must be
  80           a real-valued type such as <code class="computeroutput"><span class="keyword">float</span></code>
  81           or <code class="computeroutput"><span class="keyword">double</span></code> and not an integer
  82           type - that would overflow far too easily!
  83         </p>
  84 <p>
  85           The source code <code class="computeroutput"><span class="keyword">static_assert</span></code>
  86           and comment just after the will be:
  87         </p>
  88 <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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">);</span>
  89 <span class="comment">// factorial&lt;unsigned int&gt;(n) is not implemented</span>
  90 <span class="comment">// because it would overflow integral type T for too small n</span>
  91 <span class="comment">// to be useful. Use instead a floating-point type,</span>
  92 <span class="comment">// and convert to an unsigned type if essential, for example:</span>
  93 <span class="comment">// unsigned int nfac = static_cast&lt;unsigned int&gt;(factorial&lt;double&gt;(n));</span>
  94 <span class="comment">// See factorial documentation for more detail.</span>
  95 </pre>
  96 </td></tr>
  97 </table></div>
  98 <div class="note"><table border="0" summary="Note">
  99 <tr>
 100 <td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../doc/src/images/note.png"></td>
 101 <th align="left">Note</th>
 102 </tr>
 103 <tr><td align="left" valign="top"><p>
 104           The argument to <code class="computeroutput"><span class="identifier">double_factorial</span></code>
 105           is type <code class="computeroutput"><span class="keyword">unsigned</span></code> even though
 106           technically -1!! is defined.
 107         </p></td></tr>
 108 </table></div>
 109 <h5>
 110 <a name="math_toolkit.factorials.sf_double_factorial.h0"></a>
 111         <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>
 112       </h5>
 113 <p>
 114         The implementation uses a trivial adaptation of the factorial function, so
 115         error rates should be no more than a couple of epsilon higher.
 116       </p>
 117 <h5>
 118 <a name="math_toolkit.factorials.sf_double_factorial.h1"></a>
 119         <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>
 120       </h5>
 121 <p>
 122         The spot tests for the double factorial use data generated by functions.wolfram.com.
 123       </p>
 124 <h5>
 125 <a name="math_toolkit.factorials.sf_double_factorial.h2"></a>
 126         <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>
 127       </h5>
 128 <p>
 129         The double factorial is implemented in terms of the factorial and gamma functions
 130         using the relations:
 131       </p>
 132 <p>
 133         (2n)!! = 2<sup>n </sup> * n!
 134       </p>
 135 <p>
 136         (2n+1)!! = (2n+1)! / (2<sup>n </sup> n!)
 137       </p>
 138 <p>
 139         and
 140       </p>
 141 <p>
 142         (2n-1)!! = &#915;((2n+1)/2) * 2<sup>n </sup> / sqrt(pi)
 143       </p>
 144 </div>
 145 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 146 <td align="left"></td>
 147 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 148       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 149       Holin, Bruno Lalande, John Maddock, Johan R&#229;de, Gautam Sewani, Benjamin Sobotta,
 150       Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 151         Distributed under the Boost Software License, Version 1.0. (See accompanying
 152         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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