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| 2 | <head> | |
| 3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
| 4 | <title>Double Factorial</title> | |
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| 22 | <div class="spirit-nav"> | |
| 23 | <a accesskey="p" href="sf_factorial.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../factorials.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sf_rising_factorial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 24 | </div> | |
| 25 | <div class="section"> | |
| 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"><</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> | |
| 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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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">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> | |
| 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 15. 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 15. 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"><</span><span class="keyword">double</span><span class="special">>(</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"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">value</span><span class="special">);</span> | |
| 89 | <span class="comment">// factorial<unsigned int>(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<unsigned int>(factorial<double>(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)!! = Γ((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 © 2006-2010, 2012-2014 Nikhar Agrawal, | |
| 148 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
| 149 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
| 150 | Benjamin Sobotta, 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>) | |
| 153 | </p> | |
| 154 | </div></td> | |
| 155 | </tr></table> | |
| 156 | <hr> | |
| 157 | <div class="spirit-nav"> | |
| 158 | <a accesskey="p" href="sf_factorial.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../factorials.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="sf_rising_factorial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> | |
| 159 | </div> | |
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