mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2026-06-05 23:44:56 -04:00
Squashed 'boost/' content from commit b4feb19f2
git-subtree-dir: boost git-subtree-split: b4feb19f287ee92d87a9624b5d36b7cf46aeadeb
This commit is contained in:
Bill Somerville
@@ -0,0 +1,145 @@
|
||||
<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 2.5.1">
|
||||
<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"><</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">></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">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"><</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">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 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 the binomial coefficient: <sub>n</sub>C<sub>k</sub>.
|
||||
</p>
|
||||
<p>
|
||||
Requires k <= n.
|
||||
</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.
|
||||
</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"><</span><span class="keyword">double</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>
|
||||
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 © 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>)
|
||||
</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>
|
||||
@@ -0,0 +1,161 @@
|
||||
<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 2.5.1">
|
||||
<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"><</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>
|
||||
<td align="left"></td>
|
||||
<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>)
|
||||
</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>
|
||||
@@ -0,0 +1,193 @@
|
||||
<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 2.5.1">
|
||||
<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"><</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">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">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="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">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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</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.
|
||||
</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"><</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>
|
||||
<pre class="programlisting"><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">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">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>
|
||||
</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>
|
||||
For <code class="literal">i <= max_factorial<T>::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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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 <= max_factorial<T>::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"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></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"><</span><span class="identifier">T</span><span class="special">>::</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 © 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>)
|
||||
</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>
|
||||
@@ -0,0 +1,110 @@
|
||||
<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 2.5.1">
|
||||
<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"><</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>
|
||||
<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"><</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>
|
||||
<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 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 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 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.
|
||||
</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 © 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>)
|
||||
</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>
|
||||
@@ -0,0 +1,114 @@
|
||||
<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 2.5.1">
|
||||
<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"><</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>
|
||||
<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"><</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>
|
||||
<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 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 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) = Γ(x + i) / Γ(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 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.
|
||||
</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 © 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>)
|
||||
</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>
|
||||
Reference in New Issue
Block a user