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[polygamma] Initial docs.
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jzmaddock
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<mfenced>
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<mo>=</mo>
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Normal file
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Normal file
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|
||||
|
After Width: | Height: | Size: 4.6 KiB |
@@ -111,7 +111,7 @@ This manual is also available in <a href="http://sourceforge.net/projects/boost/
|
||||
</div>
|
||||
</div>
|
||||
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
||||
<td align="left"><p><small>Last revised: October 05, 2014 at 07:41:09 GMT</small></p></td>
|
||||
<td align="left"><p><small>Last revised: November 08, 2014 at 11:40:24 GMT</small></p></td>
|
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<td align="right"><div class="copyright-footer"></div></td>
|
||||
</tr></table>
|
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<hr>
|
||||
|
||||
@@ -24,7 +24,7 @@
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="idm1504385200"></a>Function Index</h2></div></div></div>
|
||||
<a name="idm1504580752"></a>Function Index</h2></div></div></div>
|
||||
<p><a class="link" href="s01.html#idx_id_0">A</a> <a class="link" href="s01.html#idx_id_1">B</a> <a class="link" href="s01.html#idx_id_2">C</a> <a class="link" href="s01.html#idx_id_3">D</a> <a class="link" href="s01.html#idx_id_4">E</a> <a class="link" href="s01.html#idx_id_5">F</a> <a class="link" href="s01.html#idx_id_6">G</a> <a class="link" href="s01.html#idx_id_7">H</a> <a class="link" href="s01.html#idx_id_8">I</a> <a class="link" href="s01.html#idx_id_9">J</a> <a class="link" href="s01.html#idx_id_10">K</a> <a class="link" href="s01.html#idx_id_11">L</a> <a class="link" href="s01.html#idx_id_12">M</a> <a class="link" href="s01.html#idx_id_13">N</a> <a class="link" href="s01.html#idx_id_14">O</a> <a class="link" href="s01.html#idx_id_15">P</a> <a class="link" href="s01.html#idx_id_16">Q</a> <a class="link" href="s01.html#idx_id_17">R</a> <a class="link" href="s01.html#idx_id_18">S</a> <a class="link" href="s01.html#idx_id_19">T</a> <a class="link" href="s01.html#idx_id_20">U</a> <a class="link" href="s01.html#idx_id_21">V</a> <a class="link" href="s01.html#idx_id_23">Z</a></p>
|
||||
<div class="variablelist"><dl class="variablelist">
|
||||
<dt>
|
||||
@@ -1863,6 +1863,7 @@
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/nmp.html" title="Non-Member Properties"><span class="index-entry-level-1">Non-Member Properties</span></a></p></li>
|
||||
</ul></div>
|
||||
</li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/polygamma.html" title="Polygamma"><span class="index-entry-level-0">polygamma</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/powers/powm1.html" title="powm1"><span class="index-entry-level-0">powm1</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none">
|
||||
<p><span class="index-entry-level-0">prime</span></p>
|
||||
@@ -2360,6 +2361,7 @@
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/gamma_ratios.html" title="Ratios of Gamma Functions"><span class="index-entry-level-1">Ratios of Gamma Functions</span></a></p></li>
|
||||
</ul></div>
|
||||
</li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/trigamma.html" title="Trigamma"><span class="index-entry-level-0">trigamma</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none">
|
||||
<p><span class="index-entry-level-0">trunc</span></p>
|
||||
<div class="index"><ul class="index" style="list-style-type: none; ">
|
||||
|
||||
@@ -24,7 +24,7 @@
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="idm1502500896"></a>Class Index</h2></div></div></div>
|
||||
<a name="idm1502735136"></a>Class Index</h2></div></div></div>
|
||||
<p><a class="link" href="s02.html#idx_id_25">B</a> <a class="link" href="s02.html#idx_id_26">C</a> <a class="link" href="s02.html#idx_id_27">D</a> <a class="link" href="s02.html#idx_id_28">E</a> <a class="link" href="s02.html#idx_id_29">F</a> <a class="link" href="s02.html#idx_id_30">G</a> <a class="link" href="s02.html#idx_id_31">H</a> <a class="link" href="s02.html#idx_id_32">I</a> <a class="link" href="s02.html#idx_id_35">L</a> <a class="link" href="s02.html#idx_id_36">M</a> <a class="link" href="s02.html#idx_id_37">N</a> <a class="link" href="s02.html#idx_id_38">O</a> <a class="link" href="s02.html#idx_id_39">P</a> <a class="link" href="s02.html#idx_id_40">Q</a> <a class="link" href="s02.html#idx_id_41">R</a> <a class="link" href="s02.html#idx_id_42">S</a> <a class="link" href="s02.html#idx_id_43">T</a> <a class="link" href="s02.html#idx_id_44">U</a> <a class="link" href="s02.html#idx_id_46">W</a></p>
|
||||
<div class="variablelist"><dl class="variablelist">
|
||||
<dt>
|
||||
|
||||
@@ -24,7 +24,7 @@
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="idm1502385952"></a>Typedef Index</h2></div></div></div>
|
||||
<a name="idm1502620064"></a>Typedef Index</h2></div></div></div>
|
||||
<p><a class="link" href="s03.html#idx_id_48">A</a> <a class="link" href="s03.html#idx_id_49">B</a> <a class="link" href="s03.html#idx_id_50">C</a> <a class="link" href="s03.html#idx_id_51">D</a> <a class="link" href="s03.html#idx_id_52">E</a> <a class="link" href="s03.html#idx_id_53">F</a> <a class="link" href="s03.html#idx_id_54">G</a> <a class="link" href="s03.html#idx_id_55">H</a> <a class="link" href="s03.html#idx_id_56">I</a> <a class="link" href="s03.html#idx_id_59">L</a> <a class="link" href="s03.html#idx_id_61">N</a> <a class="link" href="s03.html#idx_id_62">O</a> <a class="link" href="s03.html#idx_id_63">P</a> <a class="link" href="s03.html#idx_id_65">R</a> <a class="link" href="s03.html#idx_id_66">S</a> <a class="link" href="s03.html#idx_id_67">T</a> <a class="link" href="s03.html#idx_id_68">U</a> <a class="link" href="s03.html#idx_id_69">V</a> <a class="link" href="s03.html#idx_id_70">W</a></p>
|
||||
<div class="variablelist"><dl class="variablelist">
|
||||
<dt>
|
||||
|
||||
@@ -24,7 +24,7 @@
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="idm1502175008"></a>Macro Index</h2></div></div></div>
|
||||
<a name="idm1502409248"></a>Macro Index</h2></div></div></div>
|
||||
<p><a class="link" href="s04.html#idx_id_73">B</a> <a class="link" href="s04.html#idx_id_77">F</a></p>
|
||||
<div class="variablelist"><dl class="variablelist">
|
||||
<dt>
|
||||
|
||||
@@ -23,7 +23,7 @@
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
|
||||
<a name="idm1501833632"></a>Index</h2></div></div></div>
|
||||
<a name="idm1502002336"></a>Index</h2></div></div></div>
|
||||
<p><a class="link" href="s05.html#idx_id_96">A</a> <a class="link" href="s05.html#idx_id_97">B</a> <a class="link" href="s05.html#idx_id_98">C</a> <a class="link" href="s05.html#idx_id_99">D</a> <a class="link" href="s05.html#idx_id_100">E</a> <a class="link" href="s05.html#idx_id_101">F</a> <a class="link" href="s05.html#idx_id_102">G</a> <a class="link" href="s05.html#idx_id_103">H</a> <a class="link" href="s05.html#idx_id_104">I</a> <a class="link" href="s05.html#idx_id_105">J</a> <a class="link" href="s05.html#idx_id_106">K</a> <a class="link" href="s05.html#idx_id_107">L</a> <a class="link" href="s05.html#idx_id_108">M</a> <a class="link" href="s05.html#idx_id_109">N</a> <a class="link" href="s05.html#idx_id_110">O</a> <a class="link" href="s05.html#idx_id_111">P</a> <a class="link" href="s05.html#idx_id_112">Q</a> <a class="link" href="s05.html#idx_id_113">R</a> <a class="link" href="s05.html#idx_id_114">S</a> <a class="link" href="s05.html#idx_id_115">T</a> <a class="link" href="s05.html#idx_id_116">U</a> <a class="link" href="s05.html#idx_id_117">V</a> <a class="link" href="s05.html#idx_id_118">W</a> <a class="link" href="s05.html#idx_id_119">Z</a></p>
|
||||
<div class="variablelist"><dl class="variablelist">
|
||||
<dt>
|
||||
@@ -4256,6 +4256,8 @@
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/dist_ref/dists/weibull_dist.html" title="Weibull Distribution"><span class="index-entry-level-1">Weibull Distribution</span></a></p></li>
|
||||
</ul></div>
|
||||
</li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/polygamma.html" title="Polygamma"><span class="index-entry-level-0">Polygamma</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/polygamma.html" title="Polygamma"><span class="index-entry-level-0">polygamma</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none">
|
||||
<p><span class="index-entry-level-0">Polynomial and Rational Function Evaluation</span></p>
|
||||
<div class="index"><ul class="index" style="list-style-type: none; ">
|
||||
@@ -5196,6 +5198,8 @@
|
||||
<li class="listitem" style="list-style-type: none"><p><span class="bold"><strong><a class="link" href="../math_toolkit/dist_ref/dists/triangular_dist.html" title="Triangular Distribution"><span class="index-entry-level-1">Triangular Distribution</span></a></strong></span></p></li>
|
||||
</ul></div>
|
||||
</li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/trigamma.html" title="Trigamma"><span class="index-entry-level-0">Trigamma</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none"><p><a class="link" href="../math_toolkit/sf_gamma/trigamma.html" title="Trigamma"><span class="index-entry-level-0">trigamma</span></a></p></li>
|
||||
<li class="listitem" style="list-style-type: none">
|
||||
<p><span class="index-entry-level-0">trunc</span></p>
|
||||
<div class="index"><ul class="index" style="list-style-type: none; ">
|
||||
|
||||
@@ -27,7 +27,7 @@
|
||||
<a name="math_toolkit.conventions"></a><a class="link" href="conventions.html" title="Document Conventions">Document Conventions</a>
|
||||
</h2></div></div></div>
|
||||
<p>
|
||||
<a class="indexterm" name="idm1568087024"></a>
|
||||
<a class="indexterm" name="idm1568176912"></a>
|
||||
</p>
|
||||
<p>
|
||||
This documentation aims to use of the following naming and formatting conventions.
|
||||
|
||||
@@ -27,7 +27,7 @@
|
||||
<a name="math_toolkit.navigation"></a><a class="link" href="navigation.html" title="Navigation">Navigation</a>
|
||||
</h2></div></div></div>
|
||||
<p>
|
||||
<a class="indexterm" name="idm1568100720"></a>
|
||||
<a class="indexterm" name="idm1568190608"></a>
|
||||
</p>
|
||||
<p>
|
||||
Boost.Math documentation is provided in both HTML and PDF formats.
|
||||
|
||||
@@ -30,6 +30,8 @@
|
||||
<dt><span class="section"><a href="sf_gamma/tgamma.html">Gamma</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/lgamma.html">Log Gamma</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/digamma.html">Digamma</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/trigamma.html">Trigamma</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/polygamma.html">Polygamma</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/gamma_ratios.html">Ratios of Gamma Functions</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/igamma.html">Incomplete Gamma Functions</a></span></dt>
|
||||
<dt><span class="section"><a href="sf_gamma/igamma_inv.html">Incomplete Gamma Function
|
||||
|
||||
@@ -7,7 +7,7 @@
|
||||
<link rel="home" href="../../index.html" title="Math Toolkit 1.9.0">
|
||||
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
|
||||
<link rel="prev" href="lgamma.html" title="Log Gamma">
|
||||
<link rel="next" href="gamma_ratios.html" title="Ratios of Gamma Functions">
|
||||
<link rel="next" href="trigamma.html" title="Trigamma">
|
||||
</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<table cellpadding="2" width="100%"><tr>
|
||||
@@ -20,7 +20,7 @@
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="trigamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
@@ -62,11 +62,6 @@
|
||||
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</p>
|
||||
<p>
|
||||
There is no fully generic version of this function: all the implementations
|
||||
are tuned to specific accuracy levels, the most precise of which delivers
|
||||
34-digits of precision.
|
||||
</p>
|
||||
<p>
|
||||
The return type of this function 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 result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type
|
||||
@@ -319,6 +314,17 @@
|
||||
and BIG=20 for 128-bit reals allows the series to truncated after a suitably
|
||||
small number of terms and evaluated as a polynomial in <code class="computeroutput"><span class="number">1</span><span class="special">/(</span><span class="identifier">x</span><span class="special">*</span><span class="identifier">x</span><span class="special">)</span></code>.
|
||||
</p>
|
||||
<p>
|
||||
The arbitrary precision version of this function uses recurrence relations
|
||||
until x > BIG, and then evaluation via the asymtotic expansion above.
|
||||
As special cases integer and half integer arguments are handled via:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/digamma4.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/digamma5.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
The rational approximation <a class="link" href="../sf_implementation.html#math_toolkit.sf_implementation.rational_approximations_used">devised
|
||||
by JM</a> in the range [1,2] is derived as follows.
|
||||
@@ -379,7 +385,7 @@
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
<a accesskey="p" href="lgamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="trigamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
@@ -6,7 +6,7 @@
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.78.1">
|
||||
<link rel="home" href="../../index.html" title="Math Toolkit 1.9.0">
|
||||
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
|
||||
<link rel="prev" href="digamma.html" title="Digamma">
|
||||
<link rel="prev" href="polygamma.html" title="Polygamma">
|
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<link rel="next" href="igamma.html" title="Incomplete Gamma Functions">
|
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</head>
|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
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@@ -20,7 +20,7 @@
|
||||
</tr></table>
|
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<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
<a accesskey="p" href="polygamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
<div class="section">
|
||||
<div class="titlepage"><div><div><h3 class="title">
|
||||
@@ -349,7 +349,7 @@
|
||||
</tr></table>
|
||||
<hr>
|
||||
<div class="spirit-nav">
|
||||
<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
<a accesskey="p" href="polygamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="igamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
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</div>
|
||||
</body>
|
||||
</html>
|
||||
|
||||
269
doc/html/math_toolkit/sf_gamma/polygamma.html
Normal file
269
doc/html/math_toolkit/sf_gamma/polygamma.html
Normal file
@@ -0,0 +1,269 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
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<title>Polygamma</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
|
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<meta name="generator" content="DocBook XSL Stylesheets V1.78.1">
|
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<link rel="home" href="../../index.html" title="Math Toolkit 1.9.0">
|
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<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
|
||||
<link rel="prev" href="trigamma.html" title="Trigamma">
|
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<link rel="next" href="gamma_ratios.html" title="Ratios of Gamma Functions">
|
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|
||||
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
|
||||
<table cellpadding="2" width="100%"><tr>
|
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<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>
|
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<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="trigamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="gamma_ratios.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.sf_gamma.polygamma"></a><a class="link" href="polygamma.html" title="Polygamma">Polygamma</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.polygamma.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.synopsis"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.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">polygamma</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">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<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">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.polygamma.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.description"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
Returns the polygamma function of <span class="emphasis"><em>x</em></span>. Polygamma is defined
|
||||
as the n'th derivative of the digamma function:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/polygamma1.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/polygamma.png" align="middle"></span>
|
||||
</p>
|
||||
<p>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
||||
be used to control the behaviour of the function: how it handles errors,
|
||||
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</p>
|
||||
<p>
|
||||
The return type of this function 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 result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type
|
||||
T otherwise.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.polygamma.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.accuracy"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
The following table shows the peak errors (in units of epsilon) found on
|
||||
various platforms with various floating point types. Unless otherwise specified
|
||||
any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>.
|
||||
</p>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Significand Size
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Platform and Compiler
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Small-medium positive arguments
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Small-medium negative x
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
53
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Win32 Visual C++ 12
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=5.0 Mean=1
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=1200 Mean=65
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
64
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Win64 Mingw GCC
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=16 Mean=3
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=33 Mean=3
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
113
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Win64 Mingw GCC __float128
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=6.5 Mean=1
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=30 Mean=4
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<p>
|
||||
As shown above, error rates are generally very acceptible for moderately
|
||||
sized arguments. Error rates generally increase with increasing <span class="emphasis"><em>n</em></span>
|
||||
- this is particularly true for negative <span class="emphasis"><em>x</em></span> and even
|
||||
<span class="emphasis"><em>n</em></span> - in this situation there is a root between each negative
|
||||
integer and cancellation errors incured in computing the result increase
|
||||
with increasing <span class="emphasis"><em>n</em></span>. By the time <span class="emphasis"><em>n=170</em></span>
|
||||
the errors are so bad we can no longer even tell the sign of the result at
|
||||
<code class="computeroutput"><span class="keyword">double</span></code> precision. It should
|
||||
also be noted that for large <span class="emphasis"><em>n</em></span> the function becomes
|
||||
increasingly badly behaved when <span class="emphasis"><em>x</em></span> is negative and is
|
||||
very sensitive to slight changes in input.
|
||||
</p>
|
||||
<p>
|
||||
<span class="bold"><strong>For these reasons results should be treated with extreme
|
||||
caution when <span class="emphasis"><em>n</em></span> is large and x negative</strong></span>.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.polygamma.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.testing"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.testing">Testing</a>
|
||||
</h5>
|
||||
<p>
|
||||
Testing is against Mathematica generated spot values to 35 digit precision.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.polygamma.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.implementation"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
For x < 0 the following reflection formula is used:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/polygamma2.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
The n'th derivative of <span class="emphasis"><em>cot(x)</em></span> is tabulated for small
|
||||
<span class="emphasis"><em>n</em></span>, and for larger n has the general form:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/polygamma3.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
The coefficients of the cosine terms can be calculated iteratively starting
|
||||
from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span>: see polygamma.hpp for the full details.
|
||||
</p>
|
||||
<p>
|
||||
Once x is positive then we have two methods available to us, for small x
|
||||
we use the series expansion:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/polygamma4.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
Note that as this is an alternating series we can accelerate the series using
|
||||
Algorithm 1 from <span class="emphasis"><em>"Convergence Acceleration of Alternating
|
||||
Series", Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental
|
||||
Mathematics, 1999.</em></span> Also note that the evaluation of zeta functions
|
||||
at integer values is essentially a table lookup as <a class="link" href="../zetas/zeta.html" title="Riemann Zeta Function">zeta</a>
|
||||
is optimized for those cases.
|
||||
</p>
|
||||
<p>
|
||||
For large x we use the asymptotic expansion:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/polygamma5.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
For x in-between the two extremes we use the relation:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/polygamma6.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
to make x large enough for the asymptotic expansion to be used.
|
||||
</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, 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="trigamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="gamma_ratios.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
217
doc/html/math_toolkit/sf_gamma/trigamma.html
Normal file
217
doc/html/math_toolkit/sf_gamma/trigamma.html
Normal file
@@ -0,0 +1,217 @@
|
||||
<html>
|
||||
<head>
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
|
||||
<title>Trigamma</title>
|
||||
<link rel="stylesheet" href="../../math.css" type="text/css">
|
||||
<meta name="generator" content="DocBook XSL Stylesheets V1.78.1">
|
||||
<link rel="home" href="../../index.html" title="Math Toolkit 1.9.0">
|
||||
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
|
||||
<link rel="prev" href="digamma.html" title="Digamma">
|
||||
<link rel="next" href="polygamma.html" title="Polygamma">
|
||||
</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="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="polygamma.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.sf_gamma.trigamma"></a><a class="link" href="trigamma.html" title="Trigamma">Trigamma</a>
|
||||
</h3></div></div></div>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.trigamma.h0"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.synopsis"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.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">trigamma</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">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span>
|
||||
|
||||
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
|
||||
<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">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
|
||||
|
||||
<span class="special">}}</span> <span class="comment">// namespaces</span>
|
||||
</pre>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.trigamma.h1"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.description"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.description">Description</a>
|
||||
</h5>
|
||||
<p>
|
||||
Returns the trigamma function of <span class="emphasis"><em>x</em></span>. Trigamma is defined
|
||||
as the derivative of the digamma function:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/trigamma1.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../graphs/digamma.png" align="middle"></span>
|
||||
</p>
|
||||
<p>
|
||||
The final <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
|
||||
be used to control the behaviour of the function: how it handles errors,
|
||||
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy
|
||||
documentation for more details</a>.
|
||||
</p>
|
||||
<p>
|
||||
The return type of this function 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 result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type
|
||||
T otherwise.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.trigamma.h2"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.accuracy"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.accuracy">Accuracy</a>
|
||||
</h5>
|
||||
<p>
|
||||
The following table shows the peak errors (in units of epsilon) found on
|
||||
various platforms with various floating point types. Unless otherwise specified
|
||||
any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>.
|
||||
</p>
|
||||
<div class="informaltable"><table class="table">
|
||||
<colgroup>
|
||||
<col>
|
||||
<col>
|
||||
<col>
|
||||
</colgroup>
|
||||
<thead><tr>
|
||||
<th>
|
||||
<p>
|
||||
Significand Size
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Platform and Compiler
|
||||
</p>
|
||||
</th>
|
||||
<th>
|
||||
<p>
|
||||
Random Values
|
||||
</p>
|
||||
</th>
|
||||
</tr></thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
53
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Win32 Visual C++ 12
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=1.0 Mean=0.4
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
64
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Win64 Mingw GCC
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=1.4 Mean=0.4
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<td>
|
||||
<p>
|
||||
113
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Win64 Mingw GCC __float128
|
||||
</p>
|
||||
</td>
|
||||
<td>
|
||||
<p>
|
||||
Peak=1.0 Mean=0.5
|
||||
</p>
|
||||
</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table></div>
|
||||
<p>
|
||||
As shown above, error rates are generally very low for built in types. For
|
||||
multiprecision types, error rates are typically in the order of a few epsilon.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.trigamma.h3"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.testing"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.testing">Testing</a>
|
||||
</h5>
|
||||
<p>
|
||||
Testing is against Mathematica generated spot values to 35 digit precision.
|
||||
</p>
|
||||
<h5>
|
||||
<a name="math_toolkit.sf_gamma.trigamma.h4"></a>
|
||||
<span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.implementation"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.implementation">Implementation</a>
|
||||
</h5>
|
||||
<p>
|
||||
The arbitary precision version of this function simply calls <a class="link" href="polygamma.html" title="Polygamma">polygamma</a>.
|
||||
</p>
|
||||
<p>
|
||||
For built in fixed precision types, negative arguments are first made positive
|
||||
via:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/trigamma2.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
Then arguments in the range [0, 1) are shifted to >= 1 via:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/trigamma3.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
Then evaluation is via one of a number of rational approximations, for small
|
||||
x these are of the form:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/trigamma4.png"></span>
|
||||
</p>
|
||||
<p>
|
||||
and for large x of the form:
|
||||
</p>
|
||||
<p>
|
||||
<span class="inlinemediaobject"><img src="../../../equations/trigamma5.png"></span>
|
||||
</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, 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="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.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="polygamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
|
||||
</div>
|
||||
</body>
|
||||
</html>
|
||||
@@ -40,6 +40,8 @@
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/tgamma.html">Gamma</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/lgamma.html">Log Gamma</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/digamma.html">Digamma</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/trigamma.html">Trigamma</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/polygamma.html">Polygamma</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/gamma_ratios.html">Ratios of Gamma Functions</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma.html">Incomplete Gamma Functions</a></span></dt>
|
||||
<dt><span class="section"><a href="math_toolkit/sf_gamma/igamma_inv.html">Incomplete Gamma Function
|
||||
|
||||
@@ -99,6 +99,8 @@ and use the function's name as the link text.]
|
||||
[/gammas]
|
||||
[def __lgamma [link math_toolkit.sf_gamma.lgamma lgamma]]
|
||||
[def __digamma [link math_toolkit.sf_gamma.digamma digamma]]
|
||||
[def __trigamma [link math_toolkit.sf_gamma.trigamma trigamma]]
|
||||
[def __polygamma [link math_toolkit.sf_gamma.polygamma polygamma]]
|
||||
[def __tgamma_ratio [link math_toolkit.sf_gamma.gamma_ratios tgamma_ratio]]
|
||||
[def __tgamma_delta_ratio [link math_toolkit.sf_gamma.gamma_ratios tgamma_delta_ratio]]
|
||||
[def __tgamma [link math_toolkit.sf_gamma.tgamma tgamma]]
|
||||
@@ -464,6 +466,8 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22.
|
||||
[include sf/tgamma.qbk]
|
||||
[include sf/lgamma.qbk]
|
||||
[include sf/digamma.qbk]
|
||||
[include sf/trigamma.qbk]
|
||||
[include sf/polygamma.qbk]
|
||||
[include sf/gamma_ratios.qbk]
|
||||
[include sf/igamma.qbk]
|
||||
[include sf/igamma_inv.qbk]
|
||||
|
||||
@@ -27,10 +27,6 @@ derivative of the gamma function:
|
||||
|
||||
[optional_policy]
|
||||
|
||||
There is no fully generic version of this function: all the implementations
|
||||
are tuned to specific accuracy levels, the most precise of which delivers
|
||||
34-digits of precision.
|
||||
|
||||
The return type of this function is computed using the __arg_pomotion_rules:
|
||||
the result is of type `double` when T is an integer type, and type T otherwise.
|
||||
|
||||
@@ -97,6 +93,14 @@ Choosing BIG=10 for up to 80-bit reals, and BIG=20 for 128-bit reals allows
|
||||
the series to truncated after a suitably small number of terms and evaluated
|
||||
as a polynomial in `1/(x*x)`.
|
||||
|
||||
The arbitrary precision version of this function uses recurrence relations until
|
||||
x > BIG, and then evaluation via the asymtotic expansion above. As special cases
|
||||
integer and half integer arguments are handled via:
|
||||
|
||||
[equation digamma4]
|
||||
|
||||
[equation digamma5]
|
||||
|
||||
The rational approximation [jm_rationals] in the range [1,2] is derived as follows.
|
||||
|
||||
First a high precision approximation to digamma was constructed using a 60-term
|
||||
|
||||
104
doc/sf/polygamma.qbk
Normal file
104
doc/sf/polygamma.qbk
Normal file
@@ -0,0 +1,104 @@
|
||||
[section:polygamma Polygamma]
|
||||
|
||||
[h4 Synopsis]
|
||||
|
||||
``
|
||||
#include <boost/math/special_functions/polygamma.hpp>
|
||||
``
|
||||
|
||||
namespace boost{ namespace math{
|
||||
|
||||
template <class T>
|
||||
``__sf_result`` polygamma(int n, T z);
|
||||
|
||||
template <class T, class ``__Policy``>
|
||||
``__sf_result`` polygamma(int n, T z, const ``__Policy``&);
|
||||
|
||||
}} // namespaces
|
||||
|
||||
[h4 Description]
|
||||
|
||||
Returns the polygamma function of /x/. Polygamma is defined as the n'th
|
||||
derivative of the digamma function:
|
||||
|
||||
[equation polygamma1]
|
||||
|
||||
[graph polygamma]
|
||||
|
||||
[optional_policy]
|
||||
|
||||
The return type of this function is computed using the __arg_pomotion_rules:
|
||||
the result is of type `double` when T is an integer type, and type T otherwise.
|
||||
|
||||
[h4 Accuracy]
|
||||
|
||||
The following table shows the peak errors (in units of epsilon)
|
||||
found on various platforms with various floating point types.
|
||||
Unless otherwise specified any floating point type that is narrower
|
||||
than the one shown will have __zero_error.
|
||||
|
||||
[table
|
||||
[[Significand Size] [Platform and Compiler] [Small-medium positive arguments] [Small-medium negative x] ]
|
||||
[[53] [Win32 Visual C++ 12] [Peak=5.0 Mean=1] [Peak=1200 Mean=65]]
|
||||
[[64] [Win64 Mingw GCC] [Peak=16 Mean=3] [Peak=33 Mean=3] ]
|
||||
[[113] [Win64 Mingw GCC __float128] [Peak=6.5 Mean=1][Peak=30 Mean=4] ]
|
||||
]
|
||||
|
||||
As shown above, error rates are generally very acceptible for moderately sized
|
||||
arguments. Error rates generally increase with increasing /n/ - this is particularly true
|
||||
for negative /x/ and even /n/ - in this situation there is a root between each negative integer
|
||||
and cancellation errors incured in computing the result increase with increasing /n/. By the time
|
||||
/n=170/ the errors are so bad we can no longer even tell the sign of the result at `double` precision.
|
||||
It should also be noted that for large /n/ the function becomes increasingly badly behaved
|
||||
when /x/ is negative and is very sensitive to slight changes in input.
|
||||
|
||||
[*For these reasons results should be treated with extreme caution when /n/ is large and x negative].
|
||||
|
||||
[h4 Testing]
|
||||
|
||||
Testing is against Mathematica generated spot values to 35 digit precision.
|
||||
|
||||
[h4 Implementation]
|
||||
|
||||
For x < 0 the following reflection formula is used:
|
||||
|
||||
[equation polygamma2]
|
||||
|
||||
The n'th derivative of ['cot(x)] is tabulated for small /n/, and for larger n
|
||||
has the general form:
|
||||
|
||||
[equation polygamma3]
|
||||
|
||||
The coefficients of the cosine terms can be calculated iteratively starting
|
||||
from ['C[sub 1,0] = -1]: see polygamma.hpp for the full details.
|
||||
|
||||
Once x is positive then we have two methods available to us, for small x
|
||||
we use the series expansion:
|
||||
|
||||
[equation polygamma4]
|
||||
|
||||
Note that as this is an alternating series we can accelerate the series using
|
||||
Algorithm 1 from ['"Convergence Acceleration of Alternating Series",
|
||||
Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, Experimental Mathematics, 1999.]
|
||||
Also note that the evaluation of zeta functions at integer values is essentially a table lookup
|
||||
as __zeta is optimized for those cases.
|
||||
|
||||
For large x we use the asymptotic expansion:
|
||||
|
||||
[equation polygamma5]
|
||||
|
||||
For x in-between the two extremes we use the relation:
|
||||
|
||||
[equation polygamma6]
|
||||
|
||||
to make x large enough for the asymptotic expansion to be used.
|
||||
|
||||
[endsect][/section:polygamma The Polygamma Function]
|
||||
|
||||
[/
|
||||
Copyright 2006 John Maddock and Paul A. Bristow.
|
||||
Distributed under the Boost Software License, Version 1.0.
|
||||
(See accompanying file LICENSE_1_0.txt or copy at
|
||||
http://www.boost.org/LICENSE_1_0.txt).
|
||||
]
|
||||
|
||||
84
doc/sf/trigamma.qbk
Normal file
84
doc/sf/trigamma.qbk
Normal file
@@ -0,0 +1,84 @@
|
||||
[section:trigamma Trigamma]
|
||||
|
||||
[h4 Synopsis]
|
||||
|
||||
``
|
||||
#include <boost/math/special_functions/trigamma.hpp>
|
||||
``
|
||||
|
||||
namespace boost{ namespace math{
|
||||
|
||||
template <class T>
|
||||
``__sf_result`` trigamma(T z);
|
||||
|
||||
template <class T, class ``__Policy``>
|
||||
``__sf_result`` trigamma(T z, const ``__Policy``&);
|
||||
|
||||
}} // namespaces
|
||||
|
||||
[h4 Description]
|
||||
|
||||
Returns the trigamma function of /x/. Trigamma is defined as the
|
||||
derivative of the digamma function:
|
||||
|
||||
[equation trigamma1]
|
||||
|
||||
[graph digamma]
|
||||
|
||||
[optional_policy]
|
||||
|
||||
The return type of this function is computed using the __arg_pomotion_rules:
|
||||
the result is of type `double` when T is an integer type, and type T otherwise.
|
||||
|
||||
[h4 Accuracy]
|
||||
|
||||
The following table shows the peak errors (in units of epsilon)
|
||||
found on various platforms with various floating point types.
|
||||
Unless otherwise specified any floating point type that is narrower
|
||||
than the one shown will have __zero_error.
|
||||
|
||||
[table
|
||||
[[Significand Size] [Platform and Compiler] [Random Values] ]
|
||||
[[53] [Win32 Visual C++ 12] [Peak=1.0 Mean=0.4] ]
|
||||
[[64] [Win64 Mingw GCC] [Peak=1.4 Mean=0.4] ]
|
||||
[[113] [Win64 Mingw GCC __float128] [Peak=1.0 Mean=0.5] ]
|
||||
]
|
||||
|
||||
As shown above, error rates are generally very low for built in types.
|
||||
For multiprecision types, error rates are typically in the order of a
|
||||
few epsilon.
|
||||
|
||||
[h4 Testing]
|
||||
|
||||
Testing is against Mathematica generated spot values to 35 digit precision.
|
||||
|
||||
[h4 Implementation]
|
||||
|
||||
The arbitary precision version of this function simply calls __polygamma.
|
||||
|
||||
For built in fixed precision types, negative arguments are first made positive via:
|
||||
|
||||
[equation trigamma2]
|
||||
|
||||
Then arguments in the range \[0, 1) are shifted to >= 1 via:
|
||||
|
||||
[equation trigamma3]
|
||||
|
||||
Then evaluation is via one of a number of rational approximations, for small x these are
|
||||
of the form:
|
||||
|
||||
[equation trigamma4]
|
||||
|
||||
and for large x of the form:
|
||||
|
||||
[equation trigamma5]
|
||||
|
||||
[endsect][/section:digamma The Trigamma Function]
|
||||
|
||||
[/
|
||||
Copyright 2006 John Maddock and Paul A. Bristow.
|
||||
Distributed under the Boost Software License, Version 1.0.
|
||||
(See accompanying file LICENSE_1_0.txt or copy at
|
||||
http://www.boost.org/LICENSE_1_0.txt).
|
||||
]
|
||||
|
||||
Reference in New Issue
Block a user