[DEV] add v1.66.0

This commit is contained in:
2018-01-12 21:47:58 +01:00
parent 87059bb1af
commit a97e9ae7d4
49032 changed files with 7668950 additions and 0 deletions

View File

@@ -0,0 +1,146 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Binomial Coefficients</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients">
<link rel="prev" href="sf_falling_factorial.html" title="Falling Factorial">
<link rel="next" href="../sf_beta.html" title="Beta Functions">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="sf_falling_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_beta.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.factorials.sf_binomial"></a><a class="link" href="sf_binomial.html" title="Binomial Coefficients">Binomial Coefficients</a>
</h3></div></div></div>
<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">binomial</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">);</span>
<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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">binomial_coefficient</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">n</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">k</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Returns the binomial coefficient: <sub>n</sub>C<sub>k</sub>.
</p>
<p>
Requires k &lt;= n.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;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&#160;18.&#160;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.
</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">binomial_coefficient</span><span class="special">(</span><span class="number">10</span><span class="special">,</span> <span class="number">2</span><span class="special">);</span></code>
</p>
<p>
You will get a 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">binomial_coefficient</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</span><span class="number">10</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>
</td></tr>
</table></div>
<h5>
<a name="math_toolkit.factorials.sf_binomial.h0"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_binomial.accuracy"></a></span><a class="link" href="sf_binomial.html#math_toolkit.factorials.sf_binomial.accuracy">Accuracy</a>
</h5>
<p>
The accuracy will be the same as for the factorials for small arguments (i.e.
no more than one or two epsilon), and the <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>
function for larger arguments.
</p>
<h5>
<a name="math_toolkit.factorials.sf_binomial.h1"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_binomial.testing"></a></span><a class="link" href="sf_binomial.html#math_toolkit.factorials.sf_binomial.testing">Testing</a>
</h5>
<p>
The spot tests for the binomial coefficients use data generated by functions.wolfram.com.
</p>
<h5>
<a name="math_toolkit.factorials.sf_binomial.h2"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_binomial.implementation"></a></span><a class="link" href="sf_binomial.html#math_toolkit.factorials.sf_binomial.implementation">Implementation</a>
</h5>
<p>
Binomial coefficients are calculated using table lookup of factorials where
possible using:
</p>
<p>
<sub>n</sub>C<sub>k</sub> = n! / (k!(n-k)!)
</p>
<p>
Otherwise it is implemented in terms of the beta function using the relations:
</p>
<p>
<sub>n</sub>C<sub>k</sub> = 1 / (k * <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>(k,
n-k+1))
</p>
<p>
and
</p>
<p>
<sub>n</sub>C<sub>k</sub> = 1 / ((n-k) * <a class="link" href="../sf_beta/beta_function.html" title="Beta">beta</a>(k+1,
n-k))
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="sf_falling_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_beta.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>

View File

@@ -0,0 +1,162 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Double Factorial</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients">
<link rel="prev" href="sf_factorial.html" title="Factorial">
<link rel="next" href="sf_rising_factorial.html" title="Rising Factorial">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<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>
</div>
<div class="section">
<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">&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>
</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;18.&#160;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&#160;18.&#160;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">&lt;</span><span class="keyword">double</span><span class="special">&gt;(</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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">);</span>
<span class="comment">// factorial&lt;unsigned int&gt;(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&lt;unsigned int&gt;(factorial&lt;double&gt;(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)!! = &#915;((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>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<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>
</div>
</body>
</html>

View File

@@ -0,0 +1,194 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Factorial</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients">
<link rel="prev" href="../factorials.html" title="Factorials and Binomial Coefficients">
<link rel="next" href="sf_double_factorial.html" title="Double Factorial">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="../factorials.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_double_factorial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.factorials.sf_factorial"></a><a class="link" href="sf_factorial.html" title="Factorial">Factorial</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.factorials.sf_factorial.h0"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_factorial.synopsis"></a></span><a class="link" href="sf_factorial.html#math_toolkit.factorials.sf_factorial.synopsis">Synopsis</a>
</h5>
<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>
</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">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">&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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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>
<span class="identifier">T</span> <span class="identifier">unchecked_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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">max_factorial</span><span class="special">;</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.factorials.sf_factorial.h1"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_factorial.description"></a></span><a class="link" href="sf_factorial.html#math_toolkit.factorials.sf_factorial.description">Description</a>
</h5>
<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 below are templates where the template argument
T CANNOT 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">factorial</span><span class="special">(</span><span class="number">2</span><span class="special">);</span></code>
</p>
<p>
You will get a (perhaps perplexing) compiler error, ususally indicating
that there is no such function to be found. Instead you need to specify
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">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>
</p>
<p>
So that the return type is known.
</p>
<p>
Furthermore, 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 for quite
small values of parameter <code class="computeroutput"><span class="identifier">i</span></code>!
</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">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value</span><span class="special">);</span>
<span class="comment">// factorial&lt;unsigned int&gt;(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&lt;unsigned int&gt;(factorial&lt;double&gt;(n));</span>
<span class="comment">// See factorial documentation for more detail.</span>
</pre>
</td></tr>
</table></div>
<pre class="programlisting"><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>
<span class="identifier">T</span> <span class="identifier">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">&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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns <code class="literal">i!</code>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
For <code class="literal">i &lt;= max_factorial&lt;T&gt;::value</code> this is implemented
by table lookup, for larger values of <code class="literal">i</code>, this function
is implemented in terms of <a class="link" href="../sf_gamma/tgamma.html" title="Gamma">tgamma</a>.
</p>
<p>
If <code class="literal">i</code> is so large that the result can not be represented
in type T, then calls <a class="link" href="../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>.
</p>
<pre class="programlisting"><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>
<span class="identifier">T</span> <span class="identifier">unchecked_factorial</span><span class="special">(</span><span class="keyword">unsigned</span> <span class="identifier">i</span><span class="special">);</span>
</pre>
<p>
Returns <code class="literal">i!</code>.
</p>
<p>
Internally this function performs table lookup of the result. Further it
performs no range checking on the value of i: it is up to the caller to ensure
that <code class="literal">i &lt;= max_factorial&lt;T&gt;::value</code>. This function
is intended to be used inside inner loops that require fast table lookup
of factorials, but requires care to ensure that argument <code class="literal">i</code>
never grows too large.
</p>
<pre class="programlisting"><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>
<span class="keyword">struct</span> <span class="identifier">max_factorial</span>
<span class="special">{</span>
<span class="keyword">static</span> <span class="keyword">const</span> <span class="keyword">unsigned</span> <span class="identifier">value</span> <span class="special">=</span> <span class="identifier">X</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
This traits class defines the largest value that can be passed to <code class="literal">unchecked_factorial</code>.
The member <code class="computeroutput"><span class="identifier">value</span></code> can be used
where integral constant expressions are required: for example to define the
size of further tables that depend on the factorials.
</p>
<h5>
<a name="math_toolkit.factorials.sf_factorial.h2"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_factorial.accuracy"></a></span><a class="link" href="sf_factorial.html#math_toolkit.factorials.sf_factorial.accuracy">Accuracy</a>
</h5>
<p>
For arguments smaller than <code class="computeroutput"><span class="identifier">max_factorial</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">value</span></code> the result should be correctly rounded.
For larger arguments the accuracy will be the same as for <a class="link" href="../sf_gamma/tgamma.html" title="Gamma">tgamma</a>.
</p>
<h5>
<a name="math_toolkit.factorials.sf_factorial.h3"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_factorial.testing"></a></span><a class="link" href="sf_factorial.html#math_toolkit.factorials.sf_factorial.testing">Testing</a>
</h5>
<p>
Basic sanity checks and spot values to verify the data tables: the main tests
for the <a class="link" href="../sf_gamma/tgamma.html" title="Gamma">tgamma</a> function
handle those cases already.
</p>
<h5>
<a name="math_toolkit.factorials.sf_factorial.h4"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_factorial.implementation"></a></span><a class="link" href="sf_factorial.html#math_toolkit.factorials.sf_factorial.implementation">Implementation</a>
</h5>
<p>
The factorial function is table driven for small arguments, and is implemented
in terms of <a class="link" href="../sf_gamma/tgamma.html" title="Gamma">tgamma</a> for
larger arguments.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="../factorials.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_double_factorial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>

View File

@@ -0,0 +1,111 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Falling Factorial</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients">
<link rel="prev" href="sf_rising_factorial.html" title="Rising Factorial">
<link rel="next" href="sf_binomial.html" title="Binomial Coefficients">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="sf_rising_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_binomial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.factorials.sf_falling_factorial"></a><a class="link" href="sf_falling_factorial.html" title="Falling Factorial">Falling
Factorial</a>
</h3></div></div></div>
<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>
</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">falling_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</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">&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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">falling_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Returns the falling factorial of <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span>:
</p>
<p>
falling_factorial(x, i) = x(x-1)(x-2)(x-3)...(x-i+1)
</p>
<p>
Note that this function is only defined for positive <span class="emphasis"><em>i</em></span>,
hence the <code class="computeroutput"><span class="keyword">unsigned</span></code> second argument.
Argument <span class="emphasis"><em>x</em></span> can be either positive or negative however.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;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&#160;18.&#160;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.
</p>
<p>
The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the type of the result is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, otherwise
the type of the result is T.
</p>
<h5>
<a name="math_toolkit.factorials.sf_falling_factorial.h0"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_falling_factorial.accuracy"></a></span><a class="link" href="sf_falling_factorial.html#math_toolkit.factorials.sf_falling_factorial.accuracy">Accuracy</a>
</h5>
<p>
The accuracy will be the same as the <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>
function.
</p>
<h5>
<a name="math_toolkit.factorials.sf_falling_factorial.h1"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_falling_factorial.testing"></a></span><a class="link" href="sf_falling_factorial.html#math_toolkit.factorials.sf_falling_factorial.testing">Testing</a>
</h5>
<p>
The spot tests for the falling factorials use data generated by functions.wolfram.com.
</p>
<h5>
<a name="math_toolkit.factorials.sf_falling_factorial.h2"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_falling_factorial.implementation"></a></span><a class="link" href="sf_falling_factorial.html#math_toolkit.factorials.sf_falling_factorial.implementation">Implementation</a>
</h5>
<p>
Rising and falling factorials are implemented as ratios of gamma functions
using <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>.
Optimisations for small integer arguments are handled internally by that
function.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="sf_rising_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_binomial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>

View File

@@ -0,0 +1,115 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Rising Factorial</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="up" href="../factorials.html" title="Factorials and Binomial Coefficients">
<link rel="prev" href="sf_double_factorial.html" title="Double Factorial">
<link rel="next" href="sf_falling_factorial.html" title="Falling Factorial">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="sf_double_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_falling_factorial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.factorials.sf_rising_factorial"></a><a class="link" href="sf_rising_factorial.html" title="Rising Factorial">Rising Factorial</a>
</h3></div></div></div>
<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>
</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">);</span>
<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;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">rising_factorial</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">i</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
<p>
Returns the rising factorial of <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span>:
</p>
<p>
rising_factorial(x, i) = &#915;(x + i) / &#915;(x);
</p>
<p>
or
</p>
<p>
rising_factorial(x, i) = x(x+1)(x+2)(x+3)...(x+i-1)
</p>
<p>
Note that both <span class="emphasis"><em>x</em></span> and <span class="emphasis"><em>i</em></span> can be negative
as well as positive.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;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&#160;18.&#160;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.
</p>
<p>
The return type of these functions is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result
type calculation rules</em></span></a>: the type of the result is <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, otherwise
the type of the result is T.
</p>
<h5>
<a name="math_toolkit.factorials.sf_rising_factorial.h0"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.accuracy"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.accuracy">Accuracy</a>
</h5>
<p>
The accuracy will be the same as the <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>
function.
</p>
<h5>
<a name="math_toolkit.factorials.sf_rising_factorial.h1"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.testing"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.testing">Testing</a>
</h5>
<p>
The spot tests for the rising factorials use data generated by functions.wolfram.com.
</p>
<h5>
<a name="math_toolkit.factorials.sf_rising_factorial.h2"></a>
<span class="phrase"><a name="math_toolkit.factorials.sf_rising_factorial.implementation"></a></span><a class="link" href="sf_rising_factorial.html#math_toolkit.factorials.sf_rising_factorial.implementation">Implementation</a>
</h5>
<p>
Rising and falling factorials are implemented as ratios of gamma functions
using <a class="link" href="../sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions">tgamma_delta_ratio</a>.
Optimisations for small integer arguments are handled internally by that
function.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
</p>
</div></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="sf_double_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_falling_factorial.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>