160 lines
22 KiB
HTML
160 lines
22 KiB
HTML
<html>
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
|
|
<title>Hypergeometric pFq</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 3.0.0">
|
|
<link rel="up" href="../hypergeometric.html" title="Hypergeometric Functions">
|
|
<link rel="prev" href="hypergeometric_1f1.html" title="Hypergeometric 1F1">
|
|
<link rel="next" href="hypergeometric_refs.html" title="Hypergeometric References">
|
|
</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="hypergeometric_1f1.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../hypergeometric.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="hypergeometric_refs.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.hypergeometric.hypergeometric_pfq"></a><a class="link" href="hypergeometric_pfq.html" title="Hypergeometric pFq">Hypergeometric
|
|
<sub>p</sub>F<sub>q</sub></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">hypergeometric_pFq</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
|
|
|
|
<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">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</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">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</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">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">R</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</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">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">R</span><span class="special">>&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">R</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">R</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</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">hypergeometric_pFq</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">R</span><span class="special">>&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">R</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">p_abs_error</span> <span class="special">=</span> <span class="number">0</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
|
|
<span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Seq</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
|
|
<span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Seq</span><span class="special">&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">></span>
|
|
<span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Policy</span><span class="special">&</span> <span class="identifier">pol</span><span class="special">);</span>
|
|
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">></span>
|
|
<span class="identifier">Real</span> <span class="identifier">hypergeometric_pFq_precision</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">aj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">initializer_list</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>&</span> <span class="identifier">bj</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">Real</span><span class="special">&</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">digits10</span><span class="special">,</span> <span class="keyword">double</span> <span class="identifier">timeout</span> <span class="special">=</span> <span class="number">0.5</span><span class="special">);</span>
|
|
|
|
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
|
</pre>
|
|
<h5>
|
|
<a name="math_toolkit.hypergeometric.hypergeometric_pfq.h0"></a>
|
|
<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_pfq.description"></a></span><a class="link" href="hypergeometric_pfq.html#math_toolkit.hypergeometric.hypergeometric_pfq.description">Description</a>
|
|
</h5>
|
|
<p>
|
|
The function <code class="computeroutput"><span class="identifier">hypergeometric_pFq</span></code>
|
|
returns the result of:
|
|
</p>
|
|
<div class="blockquote"><blockquote class="blockquote"><p>
|
|
<span class="inlinemediaobject"><object type="image/svg+xml" data="../../../equations/hypergeometric_pfq_1.svg" width="409" height="53"></object></span>
|
|
</p></blockquote></div>
|
|
<p>
|
|
It is most naturally used via initializer lists as in:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">h</span> <span class="special">=</span> <span class="identifier">hypergeometric_pFq</span><span class="special">({</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">,</span><span class="number">4</span><span class="special">},</span> <span class="special">{</span><span class="number">5</span><span class="special">,</span><span class="number">6</span><span class="special">,</span><span class="number">7</span><span class="special">,</span><span class="number">8</span><span class="special">},</span> <span class="identifier">z</span><span class="special">);</span> <span class="comment">// Calculate 3F4</span>
|
|
</pre>
|
|
<p>
|
|
The optional <span class="emphasis"><em>p_abs_error</em></span> argument calculates an estimate
|
|
of the absolute error in the result from the L1 norm of the sum, plus some
|
|
other factors (see implementation below).
|
|
</p>
|
|
<p>
|
|
You should divide this value by the result to get an estimate of <span class="emphasis"><em>relative
|
|
error</em></span>.
|
|
</p>
|
|
<p>
|
|
It should be noted that the error estimates are rather crude - the error
|
|
can be significantly underestimated in some circumstances, and over-estimated
|
|
in others.
|
|
</p>
|
|
<p>
|
|
The <code class="computeroutput"><span class="identifier">hypergeometric_pFq_precision</span></code>
|
|
functions will calculate <code class="computeroutput"><span class="identifier">pFq</span></code>
|
|
to a specified number of decimal places, and if <code class="computeroutput"><span class="identifier">timeout</span></code>
|
|
is reached then they raise an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>.
|
|
Note that while these functions are defined as templates, they require type
|
|
<code class="computeroutput"><span class="identifier">Real</span></code> to be a <span class="bold"><strong>variable-precision</strong></span>
|
|
floating-point type: in practice the only type currently supported (as of
|
|
July 2019) is <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">mpfr_float</span></code>. Typical usage would be:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">mpfr_float</span> <span class="identifier">mp_type</span><span class="special">;</span>
|
|
<span class="comment">//</span>
|
|
<span class="comment">// Calculate 2F2 to 20 decimal places using a 10 second timeout:</span>
|
|
<span class="comment">//</span>
|
|
<span class="identifier">mp_type</span> <span class="identifier">result</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">hypergeometric_pFq_precision</span><span class="special">(</span>
|
|
<span class="special">{</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">a1</span><span class="special">),</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">a2</span><span class="special">)</span> <span class="special">},</span> <span class="special">{</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">b1</span><span class="special">),</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">b2</span><span class="special">)</span> <span class="special">},</span> <span class="identifier">mp_type</span><span class="special">(</span><span class="identifier">z</span><span class="special">),</span> <span class="number">20</span><span class="special">,</span> <span class="number">10.0</span>
|
|
<span class="special">);</span>
|
|
<span class="comment">//</span>
|
|
<span class="comment">// Convert the result back to a double:</span>
|
|
<span class="comment">//</span>
|
|
<span class="keyword">double</span> <span class="identifier">d_result</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">double</span><span class="special">>(</span><span class="identifier">result</span><span class="special">);</span>
|
|
</pre>
|
|
<h5>
|
|
<a name="math_toolkit.hypergeometric.hypergeometric_pfq.h1"></a>
|
|
<span class="phrase"><a name="math_toolkit.hypergeometric.hypergeometric_pfq.implementation"></a></span><a class="link" href="hypergeometric_pfq.html#math_toolkit.hypergeometric.hypergeometric_pfq.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
This function is implemented by direct summation of the series; summation
|
|
normally starts with the zeroth term, but will skip forward and sum outward
|
|
(ie in both forward and backward directions) from some term <span class="emphasis"><em>N</em></span>
|
|
when required. This is particularly important when some of the <span class="emphasis"><em>b</em></span>
|
|
arguments are negative, as in this situation the sum undergoes "false-convergence",
|
|
and then diverges again as each b<sub>j</sub> passes the origin. Consequently, were you
|
|
to plot the magnitude of the terms in the sum, you would see them pass through
|
|
a series of minima and maxima before finally converging to zero at infinity.
|
|
For some values of <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span> we can
|
|
compute where the maxima occur, and therefore sum only the terms that will
|
|
have an impact on the result. For other <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span>
|
|
values, predicting the exact locations of the maxima is not so easy, and
|
|
we simply restart summation at the point where each b<sub>j</sub> passes the origin:
|
|
this will eventually reach all the significant terms of the sum, but the
|
|
key word may well be "eventually".
|
|
</p>
|
|
<p>
|
|
The error term <span class="emphasis"><em>p_abs_error</em></span> is normally simply the sum
|
|
of the absolute values of each term multiplied by machine epsilon for type
|
|
<code class="computeroutput"><span class="identifier">Real</span></code>. However, if it was
|
|
necessary for the summation to skip forward, then <span class="emphasis"><em>p_abs_error</em></span>
|
|
is adjusted to account for the error inherent in calculating the N'th term
|
|
via logarithms.
|
|
</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 © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
|
|
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
|
|
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
|
|
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="hypergeometric_1f1.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../hypergeometric.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="hypergeometric_refs.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
|
</div>
|
|
</body>
|
|
</html>
|