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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 14. 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 14. 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 14. 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 14. 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> |
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| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
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