[DEV] add v1.76.0

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2021-10-05 21:37:46 +02:00
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commit d0115b733d
45133 changed files with 4744437 additions and 1026325 deletions

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@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Digamma</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="lgamma.html" title="Log Gamma">
<link rel="next" href="trigamma.html" title="Trigamma">
@@ -37,8 +37,8 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">digamma</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">digamma</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">digamma</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -50,16 +50,18 @@
Returns the digamma or psi function of <span class="emphasis"><em>x</em></span>. Digamma is
defined as the logarithmic derivative of the gamma function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma1.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma.svg" align="middle"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/digamma1.svg"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma.svg" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -77,7 +79,7 @@
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="table">
<a name="math_toolkit.sf_gamma.digamma.table_digamma"></a><p class="title"><b>Table&#160;6.4.&#160;Error rates for digamma</b></p>
<a name="math_toolkit.sf_gamma.digamma.table_digamma"></a><p class="title"><b>Table 8.4. Error rates for digamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for digamma">
<colgroup>
<col>
@@ -91,22 +93,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -119,25 +121,24 @@
</td>
<td>
<p>
<span class="blue">Max = 0.98&#949; (Mean = 0.369&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.84ε (Mean = 0.71ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.18ε (Mean = 0.331ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 1.84&#949; (Mean = 0.71&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1.18&#949; (Mean = 0.331&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 0.919&#949; (Mean = 0.394&#949;))
<span class="blue">Max = 1.39ε (Mean = 0.413ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.39&#949; (Mean = 0.413&#949;)</span>
<span class="blue">Max = 1.39ε (Mean = 0.413ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.39&#949; (Mean = 0.413&#949;)</span>
<span class="blue">Max = 0.98ε (Mean = 0.369ε)</span>
</p>
</td>
</tr>
@@ -149,25 +150,24 @@
</td>
<td>
<p>
<span class="blue">Max = 0.997&#949; (Mean = 0.527&#949;)</span>
<span class="blue">Max = 0.891ε (Mean = 0.0995ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 135ε (Mean = 11.9ε))<br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 2.02e+03ε (Mean = 256ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.891&#949; (Mean = 0.0995&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 135&#949; (Mean = 11.9&#949;))<br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 2.02e+03&#949; (Mean = 256&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 1.42e+04&#949; (Mean = 1.14e+03&#949;))
<span class="blue">Max = 1.37ε (Mean = 0.477ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.37&#949; (Mean = 0.477&#949;)</span>
<span class="blue">Max = 1.3 (Mean = 0.47)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.31&#949; (Mean = 0.451&#949;)</span>
<span class="blue">Max = 0.997ε (Mean = 0.527ε)</span>
</p>
</td>
</tr>
@@ -179,25 +179,24 @@
</td>
<td>
<p>
<span class="blue">Max = 0.953&#949; (Mean = 0.337&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.953ε (Mean = 0.348ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.17ε (Mean = 0.564ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 0.953&#949; (Mean = 0.348&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1.17&#949; (Mean = 0.564&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 3.5&#949; (Mean = 1.04&#949;))
<span class="blue">Max = 0.984ε (Mean = 0.361ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.984&#949; (Mean = 0.361&#949;)</span>
<span class="blue">Max = 0.984ε (Mean = 0.361ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.984&#949; (Mean = 0.361&#949;)</span>
<span class="blue">Max = 0.953ε (Mean = 0.337ε)</span>
</p>
</td>
</tr>
@@ -209,25 +208,24 @@
</td>
<td>
<p>
<span class="blue">Max = 214&#949; (Mean = 16.1&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 4.56e+04ε (Mean = 3.91e+03ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 4.6e+04ε (Mean = 3.94e+03ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 4.56e+04&#949; (Mean = 3.91e+03&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 4.6e+04&#949; (Mean = 3.94e+03&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 214&#949; (Mean = 16.4&#949;))
<span class="blue">Max = 180ε (Mean = 13ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 180&#949; (Mean = 13&#949;)</span>
<span class="blue">Max = 180ε (Mean = 13ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 180&#949; (Mean = 13&#949;)</span>
<span class="blue">Max = 214ε (Mean = 16.1ε)</span>
</p>
</td>
</tr>
@@ -239,25 +237,24 @@
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0.866ε (Mean = 0.387ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 3.58e+05ε (Mean = 1.6e+05ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 0.866&#949; (Mean = 0.387&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 3.58e+05&#949; (Mean = 1.6e+05&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 0.5&#949; (Mean = 0.224&#949;))
<span class="blue">Max = (Mean = 0.592ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.592&#949;)</span>
<span class="blue">Max = 1ε (Mean = 0.592ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.592&#949;)</span>
<span class="blue">Max = (Mean = 0ε)</span>
</p>
</td>
</tr>
@@ -269,25 +266,24 @@
</td>
<td>
<p>
<span class="blue">Max = 0.992&#949; (Mean = 0.452&#949;)</span>
<span class="blue">Max = 0.992ε (Mean = 0.215ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.18ε (Mean = 0.607ε))<br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.33ε (Mean = 0.982ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.992&#949; (Mean = 0.215&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.18&#949; (Mean = 0.607&#949;))<br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 4.33&#949; (Mean = 0.982&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> Max = 0.992&#949; (Mean = 0.383&#949;))
<span class="blue">Max = 0.888ε (Mean = 0.403ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.888&#949; (Mean = 0.403&#949;)</span>
<span class="blue">Max = 0.888ε (Mean = 0.403ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.888&#949; (Mean = 0.403&#949;)</span>
<span class="blue">Max = 0.992ε (Mean = 0.452ε)</span>
</p>
</td>
</tr>
@@ -299,25 +295,24 @@
</td>
<td>
<p>
<span class="blue">Max = 0.78&#949; (Mean = 0.314&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.09ε (Mean = 0.531ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 46.2ε (Mean = 7.24ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 1.09&#949; (Mean = 0.531&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 46.2&#949; (Mean = 7.24&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 8.56&#949; (Mean = 1.44&#949;))
<span class="blue">Max = 0.906ε (Mean = 0.409ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.906&#949; (Mean = 0.409&#949;)</span>
<span class="blue">Max = 0.906ε (Mean = 0.409ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.906&#949; (Mean = 0.409&#949;)</span>
<span class="blue">Max = 0.78ε (Mean = 0.314ε)</span>
</p>
</td>
</tr>
@@ -330,6 +325,24 @@
relative errors very close to these can be arbitrarily large, although absolute
error will remain very low.
</p>
<p>
The following error plot are based on an exhaustive search of the functions
domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/digamma____float128.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.digamma.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.digamma.testing"></a></span><a class="link" href="digamma.html#math_toolkit.sf_gamma.digamma.testing">Testing</a>
@@ -378,9 +391,10 @@
<p>
For arguments &gt; BIG the asymptotic expansion:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/digamma2.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma2.svg"></span>
</p></blockquote></div>
<p>
can be used. However, this expansion is divergent after a few terms: exactly
how many terms depends on the size of <span class="emphasis"><em>x</em></span>. Therefore the
@@ -395,12 +409,14 @@
until x &gt; BIG, and then evaluation via the asymptotic expansion above.
As special cases integer and half integer arguments are handled via:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/digamma4.svg"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/digamma5.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma4.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma5.svg"></span>
</p></blockquote></div>
<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.
@@ -410,9 +426,10 @@
differentiated <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>,
the form used is:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/digamma3.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/digamma3.svg"></span>
</p></blockquote></div>
<p>
Where P(x) and Q(x) are the polynomials from the rational form of the Lanczos
sum, and P'(x) and Q'(x) are their first derivatives. The Lanzos part of
@@ -450,11 +467,11 @@
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Derivative of the Incomplete Gamma Function</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="igamma_inv.html" title="Incomplete Gamma Function Inverses">
<link rel="next" href="../factorials.html" title="Factorials and Binomial Coefficients">
@@ -38,8 +38,8 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_derivative</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_derivative</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_derivative</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -52,13 +52,14 @@
the partial derivative with respect to <span class="emphasis"><em>x</em></span> of the incomplete
gamma function.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/derivative1.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/derivative1.svg"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -90,11 +91,11 @@
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Ratios of Gamma Functions</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="polygamma.html" title="Polygamma">
<link rel="next" href="igamma.html" title="Incomplete Gamma Functions">
@@ -33,14 +33,14 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -51,19 +51,20 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">b</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the ratio of gamma functions:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamma_ratio0.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamma_ratio0.svg"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -73,19 +74,20 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_delta_ratio</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">delta</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the ratio of gamma functions:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamma_ratio1.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamma_ratio1.svg"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -99,9 +101,10 @@
type calculation rules</em></span></a> when T1 and T2 are different types,
otherwise the result type is simple T1.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma_delta_ratio.svg" align="middle"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma_delta_ratio.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.gamma_ratios.h1"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.gamma_ratios.accuracy"></a></span><a class="link" href="gamma_ratios.html#math_toolkit.sf_gamma.gamma_ratios.accuracy">Accuracy</a>
@@ -112,7 +115,7 @@
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="table">
<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_delta_ratio"></a><p class="title"><b>Table&#160;6.7.&#160;Error rates for tgamma_delta_ratio</b></p>
<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_delta_ratio"></a><p class="title"><b>Table 8.7. Error rates for tgamma_delta_ratio</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma_delta_ratio">
<colgroup>
<col>
@@ -126,22 +129,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -154,22 +157,22 @@
</td>
<td>
<p>
<span class="blue">Max = 10.1&#949; (Mean = 1.25&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 5.83ε (Mean = 1.3ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 5.56&#949; (Mean = 0.969&#949;)</span>
<span class="blue">Max = 15.4ε (Mean = 2.09ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 15.4&#949; (Mean = 2.09&#949;)</span>
<span class="blue">Max = 7.56ε (Mean = 1.31ε)</span>
</p>
</td>
</tr>
@@ -181,22 +184,22 @@
</td>
<td>
<p>
<span class="blue">Max = 8.04&#949; (Mean = 1.31&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 7.94ε (Mean = 1.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.67&#949; (Mean = 1.29&#949;)</span>
<span class="blue">Max = 18.3ε (Mean = 2.03ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 18.3&#949; (Mean = 2.03&#949;)</span>
<span class="blue">Max = 7.43ε (Mean = 1.42ε)</span>
</p>
</td>
</tr>
@@ -208,22 +211,22 @@
</td>
<td>
<p>
<span class="blue">Max = 2.74&#949; (Mean = 0.736&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 1.96ε (Mean = 0.677ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.96&#949; (Mean = 0.677&#949;)</span>
<span class="blue">Max = 1.96ε (Mean = 0.677ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.96&#949; (Mean = 0.677&#949;)</span>
<span class="blue">Max = 2.74ε (Mean = 0.736ε)</span>
</p>
</td>
</tr>
@@ -235,22 +238,22 @@
</td>
<td>
<p>
<span class="blue">Max = 2.15&#949; (Mean = 0.685&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 1.62ε (Mean = 0.451ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.62&#949; (Mean = 0.451&#949;)</span>
<span class="blue">Max = 1.62ε (Mean = 0.451ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.62&#949; (Mean = 0.451&#949;)</span>
<span class="blue">Max = 2.15ε (Mean = 0.685ε)</span>
</p>
</td>
</tr>
@@ -262,22 +265,22 @@
</td>
<td>
<p>
<span class="blue">Max = 0.968&#949; (Mean = 0.386&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0.997ε (Mean = 0.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.997&#949; (Mean = 0.4&#949;)</span>
<span class="blue">Max = 0.997ε (Mean = 0.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.997&#949; (Mean = 0.4&#949;)</span>
<span class="blue">Max = 0.968ε (Mean = 0.386ε)</span>
</p>
</td>
</tr>
@@ -289,22 +292,22 @@
</td>
<td>
<p>
<span class="blue">Max = 0.974&#949; (Mean = 0.184&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0.853ε (Mean = 0.176ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.853&#949; (Mean = 0.176&#949;)</span>
<span class="blue">Max = 0.853ε (Mean = 0.176ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.853&#949; (Mean = 0.176&#949;)</span>
<span class="blue">Max = 0.974ε (Mean = 0.17)</span>
</p>
</td>
</tr>
@@ -312,7 +315,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_ratio"></a><p class="title"><b>Table&#160;6.8.&#160;Error rates for tgamma_ratio</b></p>
<a name="math_toolkit.sf_gamma.gamma_ratios.table_tgamma_ratio"></a><p class="title"><b>Table 8.8. Error rates for tgamma_ratio</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma_ratio">
<colgroup>
<col>
@@ -326,22 +329,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -353,22 +356,22 @@
</td>
<td>
<p>
<span class="blue">Max = 3.66&#949; (Mean = 1.27&#949;)</span>
<span class="blue">Max = 0.694ε (Mean = 0.0347ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 2.99ε (Mean = 1.15ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.09&#949; (Mean = 1.15&#949;)</span>
<span class="blue">Max = 174ε (Mean = 61.)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 174&#949; (Mean = 61.2&#949;)</span>
<span class="blue">Max = 3.28ε (Mean = 1.12ε)</span>
</p>
</td>
</tr></tbody>
@@ -381,7 +384,7 @@
<p>
Accuracy tests use data generated at very high precision (with <a href="http://shoup.net/ntl/doc/RR.txt" target="_top">NTL
RR class</a> set at 1000-bit precision: about 300 decimal digits) and
a deliberately naive calculation of &#915;(x)/&#915;(y).
a deliberately naive calculation of Γ(x)/Γ(y).
</p>
<h5>
<a name="math_toolkit.sf_gamma.gamma_ratios.h3"></a>
@@ -401,11 +404,11 @@
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Incomplete Gamma Functions</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="gamma_ratios.html" title="Ratios of Gamma Functions">
<link rel="next" href="igamma_inv.html" title="Incomplete Gamma Function Inverses">
@@ -37,26 +37,26 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -68,18 +68,18 @@
There are four <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html" target="_top">incomplete
gamma functions</a>: two are normalised versions (also known as <span class="emphasis"><em>regularized</em></span>
incomplete gamma functions) that return values in the range [0, 1], and two
are non-normalised and return values in the range [0, &#915;(a)]. Users interested
are non-normalised and return values in the range [0, Γ(a)]. Users interested
in statistical applications should use the <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html" target="_top">normalised
versions (gamma_p and gamma_q)</a>.
versions (<code class="computeroutput"><span class="identifier">gamma_p</span></code> and <code class="computeroutput"><span class="identifier">gamma_q</span></code>)</a>.
</p>
<p>
All of these functions require <span class="emphasis"><em>a &gt; 0</em></span> and <span class="emphasis"><em>z
&gt;= 0</em></span>, otherwise they return the result of <a class="link" href="../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -91,64 +91,70 @@
<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">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Policy</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<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">gamma_p</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the normalised lower incomplete gamma function of a and z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/igamma4.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma4.svg"></span>
</p></blockquote></div>
<p>
This function changes rapidly from 0 to 1 around the point z == a:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/gamma_p.svg" align="middle"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/gamma_p.svg" align="middle"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the normalised upper incomplete gamma function of a and z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/igamma3.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma3.svg"></span>
</p></blockquote></div>
<p>
This function changes rapidly from 1 to 0 around the point z == a:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/gamma_q.svg" align="middle"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/gamma_q.svg" align="middle"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma_lower</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the full (non-normalised) lower incomplete gamma function of a and
z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/igamma2.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma2.svg"></span>
</p></blockquote></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the full (non-normalised) upper incomplete gamma function of a and
z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/igamma1.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma1.svg"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.igamma.h2"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.igamma.accuracy"></a></span><a class="link" href="igamma.html#math_toolkit.sf_gamma.igamma.accuracy">Accuracy</a>
@@ -157,7 +163,7 @@
The following tables give peak and mean relative errors in over various domains
of a and z, along with comparisons to the <a href="http://www.gnu.org/software/gsl/" target="_top">GSL-1.9</a>
and <a href="http://www.netlib.org/cephes/" target="_top">Cephes</a> libraries.
Note that only results for the widest floating point type on the system are
Note that only results for the widest floating-point type on the system are
given as narrower types have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively
zero error</a>.
</p>
@@ -175,7 +181,7 @@
All values are in units of epsilon.
</p>
<div class="table">
<a name="math_toolkit.sf_gamma.igamma.table_gamma_p"></a><p class="title"><b>Table&#160;6.9.&#160;Error rates for gamma_p</b></p>
<a name="math_toolkit.sf_gamma.igamma.table_gamma_p"></a><p class="title"><b>Table 8.9. Error rates for gamma_p</b></p>
<div class="table-contents"><table class="table" summary="Error rates for gamma_p">
<colgroup>
<col>
@@ -189,22 +195,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -217,25 +223,24 @@
</td>
<td>
<p>
<span class="blue">Max = 35.1&#949; (Mean = 6.97&#949;)</span>
<span class="blue">Max = 0.955ε (Mean = 0.05ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 342ε (Mean = 45.8ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 389ε (Mean = 44ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.955&#949; (Mean = 0.05&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 342&#949; (Mean = 45.8&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 389&#949; (Mean = 44&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 492&#949; (Mean = 101&#949;))
<span class="blue">Max = 41.6ε (Mean = 8.0)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 41&#949; (Mean = 8.09&#949;)</span>
<span class="blue">Max = 239ε (Mean = 30.2ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 239&#949; (Mean = 30.2&#949;)</span>
<span class="blue">Max = 35.1ε (Mean = 6.98ε)</span>
</p>
</td>
</tr>
@@ -247,25 +252,24 @@
</td>
<td>
<p>
<span class="blue">Max = 1.54&#949; (Mean = 0.439&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 4.82ε (Mean = 0.758ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.01ε (Mean = 0.306ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 4.82&#949; (Mean = 0.758&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1.01&#949; (Mean = 0.306&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 21&#949; (Mean = 5.65&#949;))
<span class="blue">Max = (Mean = 0.464ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.461&#949;)</span>
<span class="blue">Max = 2ε (Mean = 0.461ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.472&#949;)</span>
<span class="blue">Max = 1.54ε (Mean = 0.439ε)</span>
</p>
</td>
</tr>
@@ -277,25 +281,24 @@
</td>
<td>
<p>
<span class="blue">Max = 244&#949; (Mean = 20.2&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.02e+03ε (Mean = 105ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.11e+03ε (Mean = 67.5ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 1.02e+03&#949; (Mean = 105&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1.11e+03&#949; (Mean = 67.5&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 8.18e+06&#949; (Mean = 7.69e+05&#949;))
<span class="blue">Max = 3.08e+04ε (Mean = 1.86e+03ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.08e+04&#949; (Mean = 1.86e+03&#949;)</span>
<span class="blue">Max = 3.02e+04ε (Mean = 1.91e+03ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.02e+04&#949; (Mean = 1.91e+03&#949;)</span>
<span class="blue">Max = 243ε (Mean = 20.2ε)</span>
</p>
</td>
</tr>
@@ -307,25 +310,24 @@
</td>
<td>
<p>
<span class="blue">Max = 13&#949; (Mean = 2.93&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 128ε (Mean = 22.6ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 66.2ε (Mean = 12.2ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 128&#949; (Mean = 22.6&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 66.2&#949; (Mean = 12.2&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 83.6&#949; (Mean = 22.2&#949;))
<span class="blue">Max = 11.8ε (Mean = 2.66ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 11.8&#949; (Mean = 2.65&#949;)</span>
<span class="blue">Max = 71.6ε (Mean = 9.47ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 71.6&#949; (Mean = 9.47&#949;)</span>
<span class="blue">Max = 13ε (Mean = 2.97ε)</span>
</p>
</td>
</tr>
@@ -333,7 +335,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.igamma.table_gamma_q"></a><p class="title"><b>Table&#160;6.10.&#160;Error rates for gamma_q</b></p>
<a name="math_toolkit.sf_gamma.igamma.table_gamma_q"></a><p class="title"><b>Table 8.10. Error rates for gamma_q</b></p>
<div class="table-contents"><table class="table" summary="Error rates for gamma_q">
<colgroup>
<col>
@@ -347,22 +349,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -375,25 +377,24 @@
</td>
<td>
<p>
<span class="blue">Max = 23.7&#949; (Mean = 4.03&#949;)</span>
<span class="blue">Max = 0.927ε (Mean = 0.03)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 201ε (Mean = 13.5ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 131ε (Mean = 12.7ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.927&#949; (Mean = 0.035&#949;)</span><br> <br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 201&#949; (Mean = 13.5&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 131&#949; (Mean = 12.7&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 388&#949; (Mean = 93.8&#949;))
<span class="blue">Max = 32.3ε (Mean = 6.61ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 31.3&#949; (Mean = 6.56&#949;)</span>
<span class="blue">Max = 199ε (Mean = 26.)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 199&#949; (Mean = 26.6&#949;)</span>
<span class="blue">Max = 23.7ε (Mean = 4ε)</span>
</p>
</td>
</tr>
@@ -405,26 +406,24 @@
</td>
<td>
<p>
<span class="blue">Max = 2.26&#949; (Mean = 0.732&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> <span class="red">Max = 1.38e+10ε (Mean = 1.05e+09ε))</span><br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 65.6ε (Mean = 11ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> <span class="red">Max = 1.38e+10&#949; (Mean = 1.05e+09&#949;))</span><br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 65.6&#949; (Mean = 11&#949;))<br>
(<span class="emphasis"><em>Cephes:</em></span> <span class="red">Max = 3.42e+11&#949; (Mean
= 4.1e+10&#949;))</span>
<span class="blue">Max = 2.45ε (Mean = 0.885ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.45&#949; (Mean = 0.832&#949;)</span>
<span class="blue">Max = 2.45ε (Mean = 0.819ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.25&#949; (Mean = 0.81&#949;)</span>
<span class="blue">Max = 2.2 (Mean = 0.74ε)</span>
</p>
</td>
</tr>
@@ -436,25 +435,24 @@
</td>
<td>
<p>
<span class="blue">Max = 470&#949; (Mean = 31.5&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.71e+04ε (Mean = 2.16e+03ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.02e+03ε (Mean = 62.7ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 2.71e+04&#949; (Mean = 2.16e+03&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1.02e+03&#949; (Mean = 62.7&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 8.17e+06&#949; (Mean = 7.7e+05&#949;))
<span class="blue">Max = 6.82e+03ε (Mean = 414ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 6.82e+03&#949; (Mean = 414&#949;)</span>
<span class="blue">Max = 1.15e+04ε (Mean = 733ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.15e+04&#949; (Mean = 733&#949;)</span>
<span class="blue">Max = 469ε (Mean = 31.5ε)</span>
</p>
</td>
</tr>
@@ -466,25 +464,24 @@
</td>
<td>
<p>
<span class="blue">Max = 8.48&#949; (Mean = 1.42&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 118ε (Mean = 12.5ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 138ε (Mean = 16.9ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 118&#949; (Mean = 12.5&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 138&#949; (Mean = 16.9&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 129&#949; (Mean = 26.5&#949;))
<span class="blue">Max = 11.1ε (Mean = 2.07ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 11.1&#949; (Mean = 2.09&#949;)</span>
<span class="blue">Max = 54.7ε (Mean = 6.16ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 54.7&#949; (Mean = 6.16&#949;)</span>
<span class="blue">Max = 8.72ε (Mean = 1.48ε)</span>
</p>
</td>
</tr>
@@ -492,7 +489,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.igamma.table_tgamma_lower"></a><p class="title"><b>Table&#160;6.11.&#160;Error rates for tgamma_lower</b></p>
<a name="math_toolkit.sf_gamma.igamma.table_tgamma_lower"></a><p class="title"><b>Table 8.11. Error rates for tgamma_lower</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma_lower">
<colgroup>
<col>
@@ -506,22 +503,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -534,23 +531,23 @@
</td>
<td>
<p>
<span class="blue">Max = 5.62&#949; (Mean = 1.43&#949;)</span>
<span class="blue">Max = 0.833ε (Mean = 0.0315ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 0.833ε (Mean = 0.0315ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.833&#949; (Mean = 0.0315&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 0.833&#949; (Mean = 0.0315&#949;))
<span class="blue">Max = 6.79ε (Mean = 1.46ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 6.79&#949; (Mean = 1.38&#949;)</span>
<span class="blue">Max = 363ε (Mean = 63.8ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 363&#949; (Mean = 63.8&#949;)</span>
<span class="blue">Max = 5.62ε (Mean = 1.49ε)</span>
</p>
</td>
</tr>
@@ -562,23 +559,23 @@
</td>
<td>
<p>
<span class="blue">Max = 1.57&#949; (Mean = 0.527&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 1.97ε (Mean = 0.555ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.97&#949; (Mean = 0.552&#949;)</span>
<span class="blue">Max = 1.97ε (Mean = 0.55)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.97&#949; (Mean = 0.567&#949;)</span>
<span class="blue">Max = 1.57ε (Mean = 0.525ε)</span>
</p>
</td>
</tr>
@@ -590,23 +587,23 @@
</td>
<td>
<p>
<span class="blue">Max = 2.69&#949; (Mean = 0.866&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 4.83ε (Mean = 1.15ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.83&#949; (Mean = 1.12&#949;)</span>
<span class="blue">Max = 84.7ε (Mean = 17.5ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 84.7&#949; (Mean = 17.5&#949;)</span>
<span class="blue">Max = 2.69ε (Mean = 0.849ε)</span>
</p>
</td>
</tr>
@@ -614,7 +611,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.igamma.table_tgamma_incomplete_"></a><p class="title"><b>Table&#160;6.12.&#160;Error rates for tgamma (incomplete)</b></p>
<a name="math_toolkit.sf_gamma.igamma.table_tgamma_incomplete_"></a><p class="title"><b>Table 8.12. Error rates for tgamma (incomplete)</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma (incomplete)">
<colgroup>
<col>
@@ -628,22 +625,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -656,23 +653,23 @@
</td>
<td>
<p>
<span class="blue">Max = 8.14&#949; (Mean = 1.71&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 200ε (Mean = 13.3ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 200&#949; (Mean = 13.3&#949;))
<span class="blue">Max = 8.47ε (Mean = 1.9ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.35&#949; (Mean = 1.69&#949;)</span>
<span class="blue">Max = 412ε (Mean = 95.5ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 412&#949; (Mean = 95.5&#949;)</span>
<span class="blue">Max = 8.14ε (Mean = 1.76ε)</span>
</p>
</td>
</tr>
@@ -684,24 +681,24 @@
</td>
<td>
<p>
<span class="blue">Max = 2.53&#949; (Mean = 0.66&#949;)</span>
<span class="blue">Max = 0.753ε (Mean = 0.0474ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> <span class="red">Max =
1.38e+10ε (Mean = 1.05e+09ε))</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.753&#949; (Mean = 0.0474&#949;)</span><br>
<br> (<span class="emphasis"><em>GSL 1.16:</em></span> <span class="red">Max =
1.38e+10&#949; (Mean = 1.05e+09&#949;))</span>
<span class="blue">Max = 2.31ε (Mean = 0.775ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.13&#949; (Mean = 0.717&#949;)</span>
<span class="blue">Max = 2.13ε (Mean = 0.717ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.13&#949; (Mean = 0.712&#949;)</span>
<span class="blue">Max = 2.53ε (Mean = 0.66ε)</span>
</p>
</td>
</tr>
@@ -713,23 +710,23 @@
</td>
<td>
<p>
<span class="blue">Max = 5.16&#949; (Mean = 1.44&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 117ε (Mean = 12.5ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 117&#949; (Mean = 12.5&#949;))
<span class="blue">Max = 5.52ε (Mean = 1.48ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 5.52&#949; (Mean = 1.52&#949;)</span>
<span class="blue">Max = 79.6ε (Mean = 20.9ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 79.6&#949; (Mean = 20.9&#949;)</span>
<span class="blue">Max = 5.16ε (Mean = 1.33ε)</span>
</p>
</td>
</tr>
@@ -761,33 +758,53 @@
via:
</p>
<p>
1) <span class="inlinemediaobject"><img src="../../../equations/igamma5.svg"></span>
1)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma5.svg"></span>
</p></blockquote></div>
<p>
2) <span class="inlinemediaobject"><img src="../../../equations/igamma6.svg"></span>
2)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma6.svg"></span>
</p></blockquote></div>
<p>
3) <span class="inlinemediaobject"><img src="../../../equations/igamma7.svg"></span>
3)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma7.svg"></span>
</p></blockquote></div>
<p>
The lower incomplete gamma is computed from its series representation:
</p>
<p>
4) <span class="inlinemediaobject"><img src="../../../equations/igamma8.svg"></span>
4)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma8.svg"></span>
</p></blockquote></div>
<p>
Or by subtraction of the upper integral from either &#915;(a) or 1 when <span class="emphasis"><em>x
Or by subtraction of the upper integral from either Γ(a) or 1 when <span class="emphasis"><em>x
- (1</em></span>(3x)) &gt; a and x &gt; 1.1/.
</p>
<p>
The upper integral is computed from Legendre's continued fraction representation:
</p>
<p>
5) <span class="inlinemediaobject"><img src="../../../equations/igamma9.svg"></span>
5)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma9.svg"></span>
</p></blockquote></div>
<p>
When <span class="emphasis"><em>(x &gt; 1.1)</em></span> or by subtraction of the lower integral
from either &#915;(a) or 1 when <span class="emphasis"><em>x - (1</em></span>(3x)) &lt; a/.
from either Γ(a) or 1 when <span class="emphasis"><em>x - (1</em></span>(3x)) &lt; a/.
</p>
<p>
For <span class="emphasis"><em>x &lt; 1.1</em></span> computation of the upper integral is
@@ -795,23 +812,27 @@
area. However there is another series representation for the lower integral:
</p>
<p>
6) <span class="inlinemediaobject"><img src="../../../equations/igamma10.svg"></span>
6)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma10.svg"></span>
</p></blockquote></div>
<p>
That lends itself to calculation of the upper integral via rearrangement
to:
</p>
<p>
7) <span class="inlinemediaobject"><img src="../../../equations/igamma11.svg"></span>
7)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma11.svg"></span>
</p></blockquote></div>
<p>
Refer to the documentation for <a class="link" href="../powers/powm1.html" title="powm1">powm1</a>
and <a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a> for details
of their implementation. Note however that the precision of <a class="link" href="tgamma.html" title="Gamma">tgamma1pm1</a>
is capped to either around 35 digits, or to that of the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> associated with type T - if there is one - whichever
of the two is the greater. That therefore imposes a similar limit on the
precision of this function in this region.
of their implementation.
</p>
<p>
For <span class="emphasis"><em>x &lt; 1.1</em></span> the crossover point where the result
@@ -830,15 +851,23 @@
30</em></span> then the following finite sum is used:
</p>
<p>
9) <span class="inlinemediaobject"><img src="../../../equations/igamma1f.svg"></span>
9)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma1f.svg"></span>
</p></blockquote></div>
<p>
While for half integers in the range <span class="emphasis"><em>0.5 &lt;= a &lt; 30</em></span>
While for half-integers in the range <span class="emphasis"><em>0.5 &lt;= a &lt; 30</em></span>
then the following finite sum is used:
</p>
<p>
10) <span class="inlinemediaobject"><img src="../../../equations/igamma2f.svg"></span>
10)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma2f.svg"></span>
</p></blockquote></div>
<p>
These are both more stable and more efficient than the continued fraction
alternative.
@@ -849,17 +878,33 @@
In this area an expansion due to Temme is used:
</p>
<p>
11) <span class="inlinemediaobject"><img src="../../../equations/igamma16.svg"></span>
11)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma16.svg"></span>
</p></blockquote></div>
<p>
12) <span class="inlinemediaobject"><img src="../../../equations/igamma17.svg"></span>
12)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma17.svg"></span>
</p></blockquote></div>
<p>
13) <span class="inlinemediaobject"><img src="../../../equations/igamma18.svg"></span>
13)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma18.svg"></span>
</p></blockquote></div>
<p>
14) <span class="inlinemediaobject"><img src="../../../equations/igamma19.svg"></span>
14)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma19.svg"></span>
</p></blockquote></div>
<p>
The double sum is truncated to a fixed number of terms - to give a specific
target precision - and evaluated as a polynomial-of-polynomials. There are
@@ -893,8 +938,12 @@
approximation</a> gives the greatest accuracy:
</p>
<p>
15) <span class="inlinemediaobject"><img src="../../../equations/igamma12.svg"></span>
15)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma12.svg"></span>
</p></blockquote></div>
<p>
In the event that this causes underflow/overflow then the exponent can be
reduced by a factor of <span class="emphasis"><em>a</em></span> and brought inside the power
@@ -907,8 +956,12 @@
can be avoided by using:
</p>
<p>
16) <span class="inlinemediaobject"><img src="../../../equations/igamma13.svg"></span>
16)
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/igamma13.svg"></span>
</p></blockquote></div>
<p>
when <span class="emphasis"><em>a-x</em></span> is small and a and x are large. There is still
a subtraction and therefore some cancellation errors - but the terms are
@@ -946,11 +999,11 @@
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Incomplete Gamma Function Inverses</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
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<link rel="prev" href="igamma.html" title="Incomplete Gamma Functions">
<link rel="next" href="gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">
@@ -38,26 +38,26 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -78,9 +78,9 @@
otherwise the return type is simply T1.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<div class="tip"><table border="0" summary="Tip">
@@ -92,22 +92,23 @@
<p>
When people normally talk about the inverse of the incomplete gamma function,
they are talking about inverting on parameter <span class="emphasis"><em>x</em></span>. These
are implemented here as gamma_p_inv and gamma_q_inv, and are by far the
most efficient of the inverses presented here.
are implemented here as <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
and <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code>, and are
by far the most efficient of the inverses presented here.
</p>
<p>
The inverse on the <span class="emphasis"><em>a</em></span> parameter finds use in some statistical
applications but has to be computed by rather brute force numerical techniques
and is consequently several times slower. These are implemented here as
gamma_p_inva and gamma_q_inva.
<code class="computeroutput"><span class="identifier">gamma_p_inva</span></code> and <code class="computeroutput"><span class="identifier">gamma_q_inva</span></code>.
</p>
</td></tr>
</table></div>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns a value x such that: <code class="computeroutput"><span class="identifier">q</span>
@@ -120,8 +121,8 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inv</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns a value x such that: <code class="computeroutput"><span class="identifier">p</span>
@@ -134,8 +135,8 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_q_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">q</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns a value a such that: <code class="computeroutput"><span class="identifier">q</span>
@@ -148,8 +149,8 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T1</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T2</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">gamma_p_inva</span><span class="special">(</span><span class="identifier">T1</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">T2</span> <span class="identifier">p</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns a value a such that: <code class="computeroutput"><span class="identifier">p</span>
@@ -173,7 +174,7 @@
and <a class="link" href="igamma.html" title="Incomplete Gamma Functions">gamma_q</a> functions:
</p>
<div class="table">
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv"></a><p class="title"><b>Table&#160;6.13.&#160;Error rates for gamma_p_inv</b></p>
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inv"></a><p class="title"><b>Table 8.13. Error rates for gamma_p_inv</b></p>
<div class="table-contents"><table class="table" summary="Error rates for gamma_p_inv">
<colgroup>
<col>
@@ -187,22 +188,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -215,23 +216,23 @@
</td>
<td>
<p>
<span class="blue">Max = 1.01&#949; (Mean = 0.307&#949;)</span>
<span class="blue">Max = 0.993ε (Mean = 0.15ε)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.88ε (Mean = 0.868ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.993&#949; (Mean = 0.15&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 4.88&#949; (Mean = 0.868&#949;))
<span class="blue">Max = 1.8ε (Mean = 0.406ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.62&#949; (Mean = 0.365&#949;)</span>
<span class="blue">Max = 1.89ε (Mean = 0.466ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.86&#949; (Mean = 0.405&#949;)</span>
<span class="blue">Max = 1.71ε (Mean = 0.34ε)</span>
</p>
</td>
</tr>
@@ -243,23 +244,23 @@
</td>
<td>
<p>
<span class="blue">Max = 0.924&#949; (Mean = 0.118&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0.816ε (Mean = 0.0874ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 0.816&#949; (Mean = 0.0874&#949;))
<span class="blue">Max = 0.509ε (Mean = 0.0447ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.509&#949; (Mean = 0.0447&#949;)</span>
<span class="blue">Max = 0.509ε (Mean = 0.0447ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.509&#949; (Mean = 0.0447&#949;)</span>
<span class="blue">Max = 0.924ε (Mean = 0.108ε)</span>
</p>
</td>
</tr>
@@ -271,23 +272,23 @@
</td>
<td>
<p>
<span class="blue">Max = 1.1e+003&#949; (Mean = 108&#949;)</span>
<span class="blue">Max = 441ε (Mean = 53.9ε)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 547ε (Mean = 61.6ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 441&#949; (Mean = 53.9&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 547&#949; (Mean = 61.6&#949;))
<span class="blue">Max = 9.17e+03ε (Mean = 1.45e+03ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 9.17e+03&#949; (Mean = 1.32e+03&#949;)</span>
<span class="blue">Max = 1.09e+04ε (Mean = 1.3e+03ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.09e+04&#949; (Mean = 1.46e+03&#949;)</span>
<span class="blue">Max = 1.1e+0 (Mean = 131ε)</span>
</p>
</td>
</tr>
@@ -295,7 +296,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv"></a><p class="title"><b>Table&#160;6.14.&#160;Error rates for gamma_q_inv</b></p>
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inv"></a><p class="title"><b>Table 8.14. Error rates for gamma_q_inv</b></p>
<div class="table-contents"><table class="table" summary="Error rates for gamma_q_inv">
<colgroup>
<col>
@@ -309,22 +310,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -337,23 +338,23 @@
</td>
<td>
<p>
<span class="blue">Max = 3.46&#949; (Mean = 0.475&#949;)</span>
<span class="blue">Max = 0.912ε (Mean = 0.154ε)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 4.66ε (Mean = 0.792ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.912&#949; (Mean = 0.154&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 4.66&#949; (Mean = 0.792&#949;))
<span class="blue">Max = 6.2ε (Mean = 0.627ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 6.2&#949; (Mean = 0.659&#949;)</span>
<span class="blue">Max = 6.2ε (Mean = 0.683ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 6.2&#949; (Mean = 0.661&#949;)</span>
<span class="blue">Max = 2.88ε (Mean = 0.469ε)</span>
</p>
</td>
</tr>
@@ -365,23 +366,23 @@
</td>
<td>
<p>
<span class="blue">Max = 0.814&#949; (Mean = 0.0856&#949;)</span>
<span class="blue">Max = 0.894ε (Mean = 0.0915ε)</span><br>
<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 0.894ε (Mean = 0.106ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.894&#949; (Mean = 0.0915&#949;)</span><br>
<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 0.894&#949; (Mean = 0.106&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 0.814ε (Mean = 0.0856ε)</span>
</p>
</td>
</tr>
@@ -393,23 +394,23 @@
</td>
<td>
<p>
<span class="blue">Max = 451&#949; (Mean = 65&#949;)</span>
<span class="blue">Max = 292ε (Mean = 36.4ε)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 415ε (Mean = 48.7ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 292&#949; (Mean = 36.4&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 415&#949; (Mean = 48.7&#949;))
<span class="blue">Max = 8.28e+03ε (Mean = 1.09e+03ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.28e+03&#949; (Mean = 963&#949;)</span>
<span class="blue">Max = 8.98e+03ε (Mean = 877ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 8.98e+03&#949; (Mean = 877&#949;)</span>
<span class="blue">Max = 451ε (Mean = 64.7ε)</span>
</p>
</td>
</tr>
@@ -417,7 +418,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inva"></a><p class="title"><b>Table&#160;6.15.&#160;Error rates for gamma_p_inva</b></p>
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_p_inva"></a><p class="title"><b>Table 8.15. Error rates for gamma_p_inva</b></p>
<div class="table-contents"><table class="table" summary="Error rates for gamma_p_inva">
<colgroup>
<col>
@@ -431,22 +432,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -458,29 +459,29 @@
</td>
<td>
<p>
<span class="blue">Max = 3.52&#949; (Mean = 0.997&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 7.87ε (Mean = 1.15ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 6.44&#949; (Mean = 1.1&#949;)</span>
<span class="blue">Max = 4.08ε (Mean = 1.1)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 4.08&#949; (Mean = 1.12&#949;)</span>
<span class="blue">Max = 4.92ε (Mean = 1.03ε)</span>
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inva"></a><p class="title"><b>Table&#160;6.16.&#160;Error rates for gamma_q_inva</b></p>
<a name="math_toolkit.sf_gamma.igamma_inv.table_gamma_q_inva"></a><p class="title"><b>Table 8.16. Error rates for gamma_q_inva</b></p>
<div class="table-contents"><table class="table" summary="Error rates for gamma_q_inva">
<colgroup>
<col>
@@ -494,22 +495,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -521,22 +522,22 @@
</td>
<td>
<p>
<span class="blue">Max = 5.64&#949; (Mean = 1.09&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 8.42ε (Mean = 1.3ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 6.91&#949; (Mean = 1.17&#949;)</span>
<span class="blue">Max = 7.86ε (Mean = 1.24ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.86&#949; (Mean = 1.25&#949;)</span>
<span class="blue">Max = 5.05ε (Mean = 1.08ε)</span>
</p>
</td>
</tr></tbody>
@@ -570,8 +571,8 @@
<span class="phrase"><a name="math_toolkit.sf_gamma.igamma_inv.implementation"></a></span><a class="link" href="igamma_inv.html#math_toolkit.sf_gamma.igamma_inv.implementation">Implementation</a>
</h5>
<p>
The functions gamma_p_inv and <a href="http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/" target="_top">gamma_q_inv</a>
share a common implementation.
The functions <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
and <a href="http://functions.wolfram.com/GammaBetaErf/InverseGammaRegularized/" target="_top"><code class="computeroutput"><span class="identifier">gamma_q_inv</span></code></a> share a common implementation.
</p>
<p>
First an initial approximation is computed using the methodology described
@@ -591,25 +592,28 @@
cases only one iteration is required.
</p>
<p>
The functions gamma_p_inva and gamma_q_inva also share a common implementation
but are handled separately from gamma_p_inv and gamma_q_inv.
The functions <code class="computeroutput"><span class="identifier">gamma_p_inva</span></code>
and <code class="computeroutput"><span class="identifier">gamma_q_inva</span></code> also share
a common implementation but are handled separately from <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code>
and <code class="computeroutput"><span class="identifier">gamma_q_inv</span></code>.
</p>
<p>
An initial approximation for <span class="emphasis"><em>a</em></span> is computed very crudely
so that <span class="emphasis"><em>gamma_p(a, x) ~ 0.5</em></span>, this value is then used
as a starting point for a generic derivative-free root finding algorithm.
As a consequence, these two functions are rather more expensive to compute
than the gamma_p_inv or gamma_q_inv functions. Even so, the root is usually
found in fewer than 10 iterations.
than the <code class="computeroutput"><span class="identifier">gamma_p_inv</span></code> or
<code class="computeroutput"><span class="identifier">gamma_q_inv</span></code> functions. Even
so, the root is usually found in fewer than 10 iterations.
</p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Log Gamma</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="tgamma.html" title="Gamma">
<link rel="next" href="digamma.html" title="Digamma">
@@ -37,14 +37,14 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lgamma</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lgamma</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lgamma</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<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">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">lgamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">int</span><span class="special">*</span> <span class="identifier">sign</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -56,33 +56,24 @@
The <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">lgamma function</a>
is defined by:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm1.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm1.svg"></span>
</p></blockquote></div>
<p>
The second form of the function takes a pointer to an integer, which if non-null
is set on output to the sign of tgamma(z).
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" align="middle"></span>
</p>
<p>
There are effectively two versions of this function internally: a fully generic
version that is slow, but reasonably accurate, and a much more efficient
approximation that is used where the number of digits in the significand
of T correspond to a certain <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a>. In practice, any built-in floating-point type you will
encounter has an appropriate <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> defined for it. It is also possible, given enough machine
time, to generate further <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>'s
using the program libs/math/tools/lanczos_generator.cpp.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/lgamma.svg" align="middle"></span>
</p></blockquote></div>
<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 result is of type <code class="computeroutput"><span class="keyword">double</span></code> if T is an integer type, or type T
@@ -106,7 +97,7 @@
error can be guaranteed.
</p>
<div class="table">
<a name="math_toolkit.sf_gamma.lgamma.table_lgamma"></a><p class="title"><b>Table&#160;6.3.&#160;Error rates for lgamma</b></p>
<a name="math_toolkit.sf_gamma.lgamma.table_lgamma"></a><p class="title"><b>Table 8.3. Error rates for lgamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for lgamma">
<colgroup>
<col>
@@ -120,22 +111,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -148,29 +139,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0.914&#949; (Mean = 0.167&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.958&#949; (Mean = 0.38&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 33.6ε (Mean = 2.78ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1.55ε (Mean = 0.592ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 33.6&#949; (Mean = 2.78&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1.55&#949; (Mean = 0.592&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1.55&#949; (Mean = 0.512&#949;))
<span class="blue">Max = 0.991ε (Mean = 0.308ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.67ε (Mean = 0.487ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.67ε (Mean = 0.487ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.991&#949; (Mean = 0.311&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.67&#949; (Mean = 0.487&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.67&#949; (Mean = 0.487&#949;))
<span class="blue">Max = 0.991ε (Mean = 0.383ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.36ε (Mean = 0.476ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.991&#949; (Mean = 0.383&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.36&#949; (Mean = 0.476&#949;))
<span class="blue">Max = 0.914ε (Mean = 0.175ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.958ε (Mean = 0.38ε))
</p>
</td>
</tr>
@@ -182,29 +172,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0.964&#949; (Mean = 0.462&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.962&#949; (Mean = 0.372&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 5.21ε (Mean = 1.57ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 5.21&#949; (Mean = 1.57&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 0&#949; (Mean = 0&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1.16&#949; (Mean = 0.341&#949;))
<span class="blue">Max = 1.42ε (Mean = 0.566ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.964ε (Mean = 0.543ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.964ε (Mean = 0.543ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.42&#949; (Mean = 0.566&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.964&#949; (Mean = 0.543&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.964&#949; (Mean = 0.543&#949;))
<span class="blue">Max = 1.42ε (Mean = 0.566ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.964ε (Mean = 0.543ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.42&#949; (Mean = 0.566&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.964&#949; (Mean = 0.543&#949;))
<span class="blue">Max = 0.964ε (Mean = 0.462ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.96 (Mean = 0.372ε))
</p>
</td>
</tr>
@@ -216,29 +205,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0.867&#949; (Mean = 0.468&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.906&#949; (Mean = 0.565&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 442ε (Mean = 88.8ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 7.99e+04ε (Mean = 1.68e+04ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 442&#949; (Mean = 88.8&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 7.99e+04&#949; (Mean = 1.68e+04&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1.14e+05&#949; (Mean = 2.64e+04&#949;))
<span class="blue">Max = 0.948ε (Mean = 0.36ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.615ε (Mean = 0.096ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.615ε (Mean = 0.096ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.948&#949; (Mean = 0.36&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.615&#949; (Mean = 0.096&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.615&#949; (Mean = 0.096&#949;))
<span class="blue">Max = 0.948ε (Mean = 0.36ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.71ε (Mean = 0.581ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.866&#949; (Mean = 0.355&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.71&#949; (Mean = 0.581&#949;))
<span class="blue">Max = 0.86 (Mean = 0.468ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.906ε (Mean = 0.565ε))
</p>
</td>
</tr>
@@ -250,29 +238,28 @@
</td>
<td>
<p>
<span class="blue">Max = 0.591&#949; (Mean = 0.159&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.741&#949; (Mean = 0.473&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.17e+03ε (Mean = 274ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 2.63e+05ε (Mean = 5.84e+04ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 1.17e+03&#949; (Mean = 274&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 2.63e+05&#949; (Mean = 5.84e+04&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 5.08e+05&#949; (Mean = 9.04e+04&#949;))
<span class="blue">Max = 0.878ε (Mean = 0.242ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.741ε (Mean = 0.263ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.741ε (Mean = 0.263ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.878&#949; (Mean = 0.242&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.741&#949; (Mean = 0.263&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.741&#949; (Mean = 0.263&#949;))
<span class="blue">Max = 0.878ε (Mean = 0.242ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.598ε (Mean = 0.235ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.878&#949; (Mean = 0.241&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.598&#949; (Mean = 0.235&#949;))
<span class="blue">Max = 0.591ε (Mean = 0.159ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.741ε (Mean = 0.473ε))
</p>
</td>
</tr>
@@ -284,29 +271,28 @@
</td>
<td>
<p>
<span class="blue">Max = 4.22&#949; (Mean = 1.33&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.997&#949; (Mean = 0.444&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 24.9ε (Mean = 4.6ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 4.22ε (Mean = 1.26ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 24.9&#949; (Mean = 4.6&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 2.41e+05&#949; (Mean = 4.29e+04&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 0.997&#949; (Mean = 0.429&#949;))
<span class="blue">Max = 3.81ε (Mean = 1.01ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.997ε (Mean = 0.412ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.997ε (Mean = 0.412ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.81&#949; (Mean = 1.01&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 3.01&#949; (Mean = 0.86&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.01&#949; (Mean = 0.86&#949;))
<span class="blue">Max = 3.81ε (Mean = 1.01ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.04ε (Mean = 1.01ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.81&#949; (Mean = 1.01&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.04&#949; (Mean = 1.01&#949;))
<span class="blue">Max = 4.22ε (Mean = 1.33ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.997ε (Mean = 0.444ε))
</p>
</td>
</tr>
@@ -318,36 +304,53 @@
</td>
<td>
<p>
<span class="blue">Max = 0.821&#949; (Mean = 0.419&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 249&#949; (Mean = 43.1&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 7.02ε (Mean = 1.47ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 250ε (Mean = 60.9ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 7.02&#949; (Mean = 1.47&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 4.08e+04&#949; (Mean = 7.26e+03&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1.64&#949; (Mean = 0.693&#949;))
<span class="blue">Max = 0.821ε (Mean = 0.513ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.58ε (Mean = 0.672ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.58ε (Mean = 0.672ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.821&#949; (Mean = 0.513&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.58&#949; (Mean = 0.672&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.58&#949; (Mean = 0.672&#949;))
<span class="blue">Max = 1.59ε (Mean = 0.587ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.821ε (Mean = 0.674ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.59&#949; (Mean = 0.587&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.821&#949; (Mean = 0.674&#949;))
<span class="blue">Max = 0.821ε (Mean = 0.419ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 249ε (Mean = 43.1ε))
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><h5>
<br class="table-break"><p>
The following error plot are based on an exhaustive search of the functions
domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/lgamma__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/lgamma__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/lgamma____float128.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.lgamma.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.lgamma.testing"></a></span><a class="link" href="lgamma.html#math_toolkit.sf_gamma.lgamma.testing">Testing</a>
</h5>
@@ -366,9 +369,10 @@
The generic version of this function is implemented using Sterling's approximation
for large arguments:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
</p></blockquote></div>
<p>
For small arguments, the logarithm of tgamma is used.
</p>
@@ -376,20 +380,22 @@
For negative <span class="emphasis"><em>z</em></span> the logarithm version of the reflection
formula is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm3.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm3.svg"></span>
</p></blockquote></div>
<p>
For types of known precision, the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> is used, a traits class <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">lanczos</span><span class="special">::</span><span class="identifier">lanczos_traits</span></code>
maps type T to an appropriate approximation. The logarithmic version of the
<a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a> is:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm4.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm4.svg"></span>
</p>
<p>
Where L<sub>e,g</sub> &#160; is the Lanczos sum, scaled by e<sup>g</sup>.
Where L<sub>e,g</sub> is the Lanczos sum, scaled by e<sup>g</sup>.
</p>
<p>
As before the reflection formula is used for <span class="emphasis"><em>z &lt; 0</em></span>.
@@ -410,7 +416,7 @@
</pre>
<p>
Where Y is a constant, and R(z-2) is the rational approximation: optimised
so that it's absolute error is tiny compared to Y. In addition small values
so that its absolute error is tiny compared to Y. In addition, small values
of z greater than 3 can handled by argument reduction using the recurrence
relation:
</p>
@@ -454,27 +460,29 @@
also be fed into log1p. Crucially, all of the terms tend to zero, as <span class="emphasis"><em>z
-&gt; 1</em></span>:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm5.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm5.svg"></span>
</p>
<p>
The C<sub>k</sub> &#160; terms in the above are the same as in the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
The C<sub>k</sub> terms in the above are the same as in the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a>.
</p>
<p>
A similar rearrangement can be performed at <span class="emphasis"><em>z = 2</em></span>:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm6.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/lgamm6.svg"></span>
</p></blockquote></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Polygamma</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="trigamma.html" title="Trigamma">
<link rel="next" href="gamma_ratios.html" title="Ratios of Gamma Functions">
@@ -37,8 +37,8 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -50,21 +50,26 @@
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.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma1.svg"></span>
</p></blockquote></div>
<p>
The following graphs illustrate the behaviour of the function for odd and
even order:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span>
<span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -82,7 +87,7 @@
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="table">
<a name="math_toolkit.sf_gamma.polygamma.table_polygamma"></a><p class="title"><b>Table&#160;6.6.&#160;Error rates for polygamma</b></p>
<a name="math_toolkit.sf_gamma.polygamma.table_polygamma"></a><p class="title"><b>Table 8.6. Error rates for polygamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for polygamma">
<colgroup>
<col>
@@ -96,22 +101,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -124,24 +129,24 @@
</td>
<td>
<p>
<span class="blue">Max = 6.34&#949; (Mean = 1.53&#949;)</span>
<span class="blue">Max = 0.824ε (Mean = 0.0574ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 62.9ε (Mean = 12.8ε))<br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 108ε (Mean = 15.2ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.824&#949; (Mean = 0.0574&#949;)</span><br>
<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 108&#949; (Mean = 15.2&#949;))<br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 62.9&#949; (Mean = 12.8&#949;))
<span class="blue">Max = 7.38ε (Mean = 1.84ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 7.38&#949; (Mean = 1.84&#949;)</span>
<span class="blue">Max = 34.3ε (Mean = 7.65ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 18.3&#949; (Mean = 4.16&#949;)</span>
<span class="blue">Max = 9.32ε (Mean = 1.95ε)</span>
</p>
</td>
</tr>
@@ -153,27 +158,26 @@
</td>
<td>
<p>
<span class="blue">Max = 150&#949; (Mean = 15.1&#949;)</span>
<span class="blue">Max = 0.998ε (Mean = 0.0592ε)</span><br>
<br> (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 244ε (Mean = 32.8ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_large_arguments">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
<span class="red">Max = 1.71e+56ε (Mean = 1.01e+55ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_large_arguments">And
other failures.</a>)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.998&#949; (Mean = 0.0592&#949;)</span><br>
<br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max
= 1.71e+56&#949; (Mean = 1.01e+55&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_large_arguments">And
other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span>
Max = 244&#949; (Mean = 32.8&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_large_arguments">And
other failures.</a>)
<span class="blue">Max = 2.23ε (Mean = 0.323ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.23&#949; (Mean = 0.323&#949;)</span>
<span class="blue">Max = 11.1ε (Mean = 0.848ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.35&#949; (Mean = 0.34&#949;)</span>
<span class="blue">Max = 150ε (Mean = 13.9ε)</span>
</p>
</td>
</tr>
@@ -185,27 +189,26 @@
</td>
<td>
<p>
<span class="blue">Max = 497&#949; (Mean = 129&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.516&#949; (Mean = 0.022&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max = +INF&#949; (Mean
= +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_negative_arguments">And
other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span>
Max = 36.6&#949; (Mean = 3.04&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_negative_arguments">And
<span class="blue">Max = 0.516ε (Mean = 0.022ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 36.6ε (Mean = 3.04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_negative_arguments">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_negative_arguments">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 269&#949; (Mean = 87.7&#949;)</span>
<span class="blue">Max = 269ε (Mean = 87.7ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 269&#949; (Mean = 87.9&#949;)</span>
<span class="blue">Max = 269ε (Mean = 88.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 497ε (Mean = 129ε)</span>
</p>
</td>
</tr>
@@ -217,26 +220,26 @@
</td>
<td>
<p>
<span class="blue">Max = 162&#949; (Mean = 101&#949;)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_large_negative_arguments">And
other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span>
Max = 1.79&#949; (Mean = 0.197&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_large_negative_arguments">And
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.79ε (Mean = 0.197ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_large_negative_arguments">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
Max = 0ε (Mean = 0ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_large_negative_arguments">And
other failures.</a>)
</p>
</td>
<td>
<p>
<span class="blue">Max = 155&#949; (Mean = 96.4&#949;)</span>
<span class="blue">Max = 155ε (Mean = 96.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 155&#949; (Mean = 96.4&#949;)</span>
<span class="blue">Max = 155ε (Mean = 96.4ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 162ε (Mean = 101ε)</span>
</p>
</td>
</tr>
@@ -248,24 +251,24 @@
</td>
<td>
<p>
<span class="blue">Max = 3&#949; (Mean = 0.496&#949;)</span>
<span class="blue">Max = (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 15.2ε (Mean = 5.03ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 106ε (Mean = 20ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 106&#949; (Mean = 20&#949;))<br> (<span class="emphasis"><em>GSL 1.16:</em></span>
Max = 15.2&#949; (Mean = 5.03&#949;))
<span class="blue">Max = 3.33ε (Mean = 0.75ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.33&#949; (Mean = 0.75&#949;)</span>
<span class="blue">Max = 3.33ε (Mean = 0.75ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.33&#949; (Mean = 0.75&#949;)</span>
<span class="blue">Max = 3ε (Mean = 0.496ε)</span>
</p>
</td>
</tr>
@@ -277,26 +280,26 @@
</td>
<td>
<p>
<span class="blue">Max = 200&#949; (Mean = 57.2&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 151ε (Mean = 39.3ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_GSL_2_1_Mathematica_Data_Large_orders_and_other_bug_cases">And
other failures.</a>)<br> (<span class="emphasis"><em>Rmath 3.2.3:</em></span>
<span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_7_1_0_linux_double_polygamma_Rmath_3_2_3_Mathematica_Data_Large_orders_and_other_bug_cases">And
other failures.</a>)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> <span class="red">Max = +INF&#949; (Mean = +INF&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_Large_orders_and_other_bug_cases">And
other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span>
Max = 151&#949; (Mean = 39.3&#949;) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_Large_orders_and_other_bug_cases">And
other failures.</a>)
<span class="blue">Max = 54.5ε (Mean = 13.3ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 54.5&#949; (Mean = 13.3&#949;)</span>
<span class="blue">Max = 145ε (Mean = 55.9ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 90.1&#949; (Mean = 30.6&#949;)</span>
<span class="blue">Max = 200ε (Mean = 57.2ε)</span>
</p>
</td>
</tr>
@@ -308,7 +311,7 @@
sized arguments. Error rates should stay low for exact inputs, however, please
note that the function becomes exceptionally sensitive to small changes in
input for large n and negative x, indeed for cases where <span class="emphasis"><em>n!</em></span>
would overflow, the function changes directly from -&#8734; to +&#8734; somewhere between
would overflow, the function changes directly from -∞ to +∞ somewhere between
each negative integer - <span class="emphasis"><em>these cases are not handled correctly</em></span>.
</p>
<p>
@@ -329,23 +332,26 @@
<p>
For x &lt; 0 the following reflection formula is used:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span>
</p></blockquote></div>
<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.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma3.svg"></span>
</p></blockquote></div>
<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> and then using
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span>
</p></blockquote></div>
<p>
to generate coefficients for n+1.
</p>
@@ -357,9 +363,10 @@
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.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma4.svg"></span>
</p></blockquote></div>
<p>
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
@@ -368,35 +375,39 @@
<p>
For large x we use the asymptotic expansion:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span>
</p></blockquote></div>
<p>
For x in-between the two extremes we use the relation:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span>
</p></blockquote></div>
<p>
to make x large enough for the asymptotic expansion to be used.
</p>
<p>
There are also two special cases:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span>
</p></blockquote></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Gamma</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<link rel="up" href="../sf_gamma.html" title="Gamma Functions">
<link rel="prev" href="../sf_gamma.html" title="Gamma Functions">
<link rel="next" href="lgamma.html" title="Log Gamma">
@@ -37,14 +37,14 @@
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -55,35 +55,26 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<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">tgamma</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma</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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns the "true gamma" (hence name tgamma) of value z:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamm1.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm1.svg"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma.svg" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
There are effectively two versions of the <a href="http://en.wikipedia.org/wiki/Gamma_function" target="_top">tgamma</a>
function internally: a fully generic version that is slow, but reasonably
accurate, and a much more efficient approximation that is used where the
number of digits in the significand of T correspond to a certain <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a>. In practice any built in floating point type you will
encounter has an appropriate <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> defined for it. It is also possible, given enough machine
time, to generate further <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos approximation</a>'s
using the program libs/math/tools/lanczos_generator.cpp.
</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 <code class="computeroutput"><span class="keyword">double</span></code>
@@ -92,16 +83,14 @@
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<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">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">tgamma1pm1</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">dz</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
</pre>
<p>
Returns <code class="computeroutput"><span class="identifier">tgamma</span><span class="special">(</span><span class="identifier">dz</span> <span class="special">+</span> <span class="number">1</span><span class="special">)</span> <span class="special">-</span> <span class="number">1</span></code>.
Internally the implementation does not make use of the addition and subtraction
implied by the definition, leading to accurate results even for very small
<code class="computeroutput"><span class="identifier">dz</span></code>. However, the implementation
is capped to either 35 digit accuracy, or to the precision of the <a class="link" href="../lanczos.html" title="The Lanczos Approximation">Lanczos
approximation</a> associated with type T, whichever is more accurate.
<code class="computeroutput"><span class="identifier">dz</span></code>.
</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
@@ -109,9 +98,9 @@
when T is an integer type, and T otherwise.
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<h5>
@@ -126,7 +115,7 @@
zero error</a>.
</p>
<div class="table">
<a name="math_toolkit.sf_gamma.tgamma.table_tgamma"></a><p class="title"><b>Table&#160;6.1.&#160;Error rates for tgamma</b></p>
<a name="math_toolkit.sf_gamma.tgamma.table_tgamma"></a><p class="title"><b>Table 8.1. Error rates for tgamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma">
<colgroup>
<col>
@@ -140,22 +129,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -168,29 +157,28 @@
</td>
<td>
<p>
<span class="blue">Max = 1.85&#949; (Mean = 0.491&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.17&#949; (Mean = 0.928&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 3.95ε (Mean = 0.783ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 314ε (Mean = 93.4ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 3.95&#949; (Mean = 0.783&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 314&#949; (Mean = 93.4&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 3.19&#949; (Mean = 0.884&#949;))
<span class="blue">Max = 2.67ε (Mean = 0.617ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.66ε (Mean = 0.584ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.66ε (Mean = 0.584ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.96&#949; (Mean = 0.483&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.66&#949; (Mean = 0.584&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.66&#949; (Mean = 0.584&#949;))
<span class="blue">Max = 172ε (Mean = 41ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 172&#949; (Mean = 41&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 1.85ε (Mean = 0.566ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.17ε (Mean = 0.928ε))
</p>
</td>
</tr>
@@ -202,29 +190,28 @@
</td>
<td>
<p>
<span class="blue">Max = 1.96&#949; (Mean = 0.684&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.405&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 4.51ε (Mean = 1.92ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1ε (Mean = 0.335ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 4.51&#949; (Mean = 1.92&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1&#949; (Mean = 0.335&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1&#949; (Mean = 0.548&#949;))
<span class="blue">Max = (Mean = 0.608ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1ε (Mean = 0.376ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1ε (Mean = 0.376ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.73&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1&#949; (Mean = 0.376&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.376&#949;))
<span class="blue">Max = 2ε (Mean = 0.647ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.5ε (Mean = 0.0791ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.647&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.5&#949; (Mean = 0.0791&#949;))
<span class="blue">Max = 1.5ε (Mean = 0.635ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1ε (Mean = 0.405ε))
</p>
</td>
</tr>
@@ -236,29 +223,28 @@
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.865&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.4&#949;))
<span class="blue">Max = (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 4.41ε (Mean = 1.81ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1ε (Mean = 0.32ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 4.41&#949; (Mean = 1.81&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1&#949; (Mean = 0.32&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1&#949; (Mean = 0.518&#949;))
<span class="blue">Max = 2.51ε (Mean = 1.02ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.918ε (Mean = 0.203ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.918ε (Mean = 0.203ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.85&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.918&#949; (Mean = 0.203&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.918&#949; (Mean = 0.203&#949;))
<span class="blue">Max = 3.01ε (Mean = 1.06ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1ε (Mean = 0.175ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.01&#949; (Mean = 1.06&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1&#949; (Mean = 0.175&#949;))
<span class="blue">Max = 1.1ε (Mean = 0.59ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1ε (Mean = 0.))
</p>
</td>
</tr>
@@ -270,29 +256,28 @@
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.995&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 7.95ε (Mean = 3.12ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 1ε (Mean = 0.191ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 7.95&#949; (Mean = 3.12&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 1&#949; (Mean = 0.191&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 1.09&#949; (Mean = 0.502&#949;))
<span class="blue">Max = 4.1ε (Mean = 1.55ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 0.558ε (Mean = 0.298ε))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.558ε (Mean = 0.298ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2&#949; (Mean = 0.913&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 0.558&#949; (Mean = 0.298&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.558&#949; (Mean = 0.298&#949;))
<span class="blue">Max = 5.01ε (Mean = 1.89ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 5.01&#949; (Mean = 1.89&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 2ε (Mean = 0.733ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
</tr>
@@ -304,29 +289,28 @@
</td>
<td>
<p>
<span class="blue">Max = 1.73&#949; (Mean = 0.729&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.866&#949; (Mean = 0.445&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 2.6ε (Mean = 1.05ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 34.9ε (Mean = 9.2ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 2.6&#949; (Mean = 1.05&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 6.34e+05&#949; (Mean = 1.2e+05&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 2.6&#949; (Mean = 0.956&#949;))
<span class="blue">Max = 1.75ε (Mean = 0.895ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 2.26ε (Mean = 1.0))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.26ε (Mean = 1.08ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 2.6&#949; (Mean = 0.985&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 2.26&#949; (Mean = 1.08&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 2.26&#949; (Mean = 1.08&#949;))
<span class="blue">Max = 1.75ε (Mean = 0.819ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.75&#949; (Mean = 0.819&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 1.86ε (Mean = 0.881ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0.866ε (Mean = 0.445ε))
</p>
</td>
</tr>
@@ -338,29 +322,28 @@
</td>
<td>
<p>
<span class="blue">Max = 1.8&#949; (Mean = 0.817&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.87e+004&#949; (Mean = 6.71e+003&#949;))
<span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>GSL
2.1:</em></span> Max = 1.8ε (Mean = 0.782ε))<br> (<span class="emphasis"><em>Rmath
3.2.3:</em></span> Max = 3.89e+04ε (Mean = 9.52e+03ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span><br> <br> (<span class="emphasis"><em>GSL
1.16:</em></span> Max = 1.8&#949; (Mean = 0.782&#949;))<br> (<span class="emphasis"><em>Rmath
3.0.2:</em></span> Max = 6.36e+06&#949; (Mean = 1.13e+06&#949;))<br> (<span class="emphasis"><em>Cephes:</em></span>
Max = 2.7&#949; (Mean = 0.988&#949;))
<span class="blue">Max = 2.69ε (Mean = 1.09ε)</span><br> <br>
(<span class="emphasis"><em>&lt;cmath&gt;:</em></span> Max = 1.79ε (Mean = 0.7))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.79ε (Mean = 0.75ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.8&#949; (Mean = 0.847&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;tr1/cmath&gt;:</em></span> Max = 1.79&#949; (Mean = 0.75&#949;))<br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 1.79&#949; (Mean = 0.75&#949;))
<span class="blue">Max = 98.5ε (Mean = 53.4ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0ε (Mean = 0ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 98.5&#949; (Mean = 53.4&#949;)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 0&#949; (Mean = 0&#949;))
<span class="blue">Max = 2.7ε (Mean = 1.35ε)</span><br> <br>
(<span class="emphasis"><em>&lt;math.h&gt;:</em></span> Max = 3.87e+04ε (Mean = 6.71e+03ε))
</p>
</td>
</tr>
@@ -368,7 +351,7 @@
</table></div>
</div>
<br class="table-break"><div class="table">
<a name="math_toolkit.sf_gamma.tgamma.table_tgamma1pm1"></a><p class="title"><b>Table&#160;6.2.&#160;Error rates for tgamma1pm1</b></p>
<a name="math_toolkit.sf_gamma.tgamma.table_tgamma1pm1"></a><p class="title"><b>Table 8.2. Error rates for tgamma1pm1</b></p>
<div class="table-contents"><table class="table" summary="Error rates for tgamma1pm1">
<colgroup>
<col>
@@ -382,22 +365,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -409,28 +392,46 @@
</td>
<td>
<p>
<span class="blue">Max = 0.982&#949; (Mean = 0.399&#949;)</span>
<span class="blue">Max = 0ε (Mean = 0ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 0&#949; (Mean = 0&#949;)</span>
<span class="blue">Max = 1.12ε (Mean = 0.49ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.12&#949; (Mean = 0.49&#949;)</span>
<span class="blue">Max = 6.61ε (Mean = 0.84ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 3.97&#949; (Mean = 0.713&#949;)</span>
<span class="blue">Max = 3.31ε (Mean = 0.517ε)</span>
</p>
</td>
</tr></tbody>
</table></div>
</div>
<br class="table-break"><h5>
<br class="table-break"><p>
The following error plot are based on an exhaustive search of the functions
domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/tgamma____float128.svg" align="middle"></span>
</p></blockquote></div>
<h5>
<a name="math_toolkit.sf_gamma.tgamma.h3"></a>
<span class="phrase"><a name="math_toolkit.sf_gamma.tgamma.testing"></a></span><a class="link" href="tgamma.html#math_toolkit.sf_gamma.tgamma.testing">Testing</a>
</h5>
@@ -451,11 +452,13 @@
</h5>
<p>
The generic version of the <code class="computeroutput"><span class="identifier">tgamma</span></code>
function is implemented Sterling's approximation for lgamma for large z:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
function is implemented Sterling's approximation for <code class="computeroutput"><span class="identifier">lgamma</span></code>
for large z:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamma6.svg"></span>
</p></blockquote></div>
<p>
Following exponentiation, downward recursion is then used for small values
of z.
@@ -469,25 +472,28 @@
For z in the range -20 &lt; z &lt; 1 then recursion is used to shift to z
&gt; 1 via:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm3.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamm3.svg"></span>
</p></blockquote></div>
<p>
For very small z, this helps to preserve the identity:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm4.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamm4.svg"></span>
</p></blockquote></div>
<p>
For z &lt; -20 the reflection formula:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/gamm5.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/gamm5.svg"></span>
</p>
<p>
is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(&#960; &#160; *
is used. Particular care has to be taken to evaluate the <code class="literal">z * sin(π *
z)</code> part: a special routine is used to reduce z prior to multiplying
by &#960; &#160; to ensure that the result in is the range [0, &#960;/2]. Without this an excessive
by π to ensure that the result in is the range [0, π/2]. Without this an excessive
amount of error occurs in this region (which is hard enough already, as the
rate of change near a negative pole is <span class="emphasis"><em>exceptionally</em></span>
high).
@@ -514,11 +520,11 @@
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>

View File

@@ -1,10 +1,10 @@
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Trigamma</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.6.0">
<link rel="home" href="../../index.html" title="Math Toolkit 3.0.0">
<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">
@@ -35,10 +35,10 @@
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</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">x</span><span class="special">);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">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&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
<span class="special">}}</span> <span class="comment">// namespaces</span>
</pre>
@@ -50,16 +50,18 @@
Returns the trigamma function of <span class="emphasis"><em>x</em></span>. Trigamma is defined
as the derivative of the digamma function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span>
</p>
<p>
<span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span>
</p>
<p>
The final <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
be used to control the behaviour of the function: how it handles errors,
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;18.&#160;Policies: Controlling Precision, Error Handling etc">policy
what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy
documentation for more details</a>.
</p>
<p>
@@ -77,7 +79,7 @@
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="table">
<a name="math_toolkit.sf_gamma.trigamma.table_trigamma"></a><p class="title"><b>Table&#160;6.5.&#160;Error rates for trigamma</b></p>
<a name="math_toolkit.sf_gamma.trigamma.table_trigamma"></a><p class="title"><b>Table 8.5. Error rates for trigamma</b></p>
<div class="table-contents"><table class="table" summary="Error rates for trigamma">
<colgroup>
<col>
@@ -91,22 +93,22 @@
</th>
<th>
<p>
Microsoft Visual C++ version 12.0<br> Win32<br> double
GNU C++ version 7.1.0<br> linux<br> double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> double
GNU C++ version 7.1.0<br> linux<br> long double
</p>
</th>
<th>
<p>
GNU C++ version 5.1.0<br> linux<br> long double
Sun compiler version 0x5150<br> Sun Solaris<br> long double
</p>
</th>
<th>
<p>
Sun compiler version 0x5130<br> Sun Solaris<br> long double
Microsoft Visual C++ version 14.1<br> Win32<br> double
</p>
</th>
</tr></thead>
@@ -118,24 +120,24 @@
</td>
<td>
<p>
<span class="blue">Max = 1&#949; (Mean = 0.382&#949;)</span>
<span class="blue">Max = 0.998ε (Mean = 0.105ε)</span><br> <br>
(<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.34e+04ε (Mean = 1.49e+03ε))<br>
(<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.34e+04ε (Mean = 1.51e+03ε))
</p>
</td>
<td>
<p>
<span class="blue">Max = 0.998&#949; (Mean = 0.105&#949;)</span><br> <br>
(<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.34e+04&#949; (Mean = 1.51e+03&#949;))<br>
(<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.34e+04&#949; (Mean = 1.49e+03&#949;))
<span class="blue">Max = 1.28ε (Mean = 0.449ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.28&#949; (Mean = 0.449&#949;)</span>
<span class="blue">Max = 1.28ε (Mean = 0.449ε)</span>
</p>
</td>
<td>
<p>
<span class="blue">Max = 1.28&#949; (Mean = 0.447&#949;)</span>
<span class="blue">Max = 1ε (Mean = 0.382ε)</span>
</p>
</td>
</tr></tbody>
@@ -145,6 +147,24 @@
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>
<p>
The following error plot are based on an exhaustive search of the functions
domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code>
precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span>
<span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>.
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/trigamma__double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/trigamma__80_bit_long_double.svg" align="middle"></span>
</p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../graphs/trigamma____float128.svg" align="middle"></span>
</p></blockquote></div>
<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>
@@ -160,39 +180,43 @@
The arbitrary 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
For built in fixed-precision types, negative arguments are first made positive
via:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span>
</p></blockquote></div>
<p>
Then arguments in the range [0, 1) are shifted to &gt;= 1 via:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span>
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span>
</p></blockquote></div>
<p>
Then evaluation is via one of a number of rational approximations, for small
x these are of the form:
<span class="emphasis"><em>x</em></span> these are of the form:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span>
</p></blockquote></div>
<p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span>
</p>
<p>
and for large x of the form:
</p>
<p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span>
and for large <span class="emphasis"><em>x</em></span> of the form:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span>
</p></blockquote></div>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014, 2017 Nikhar
Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam
Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle Walker
and Xiaogang Zhang<p>
<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>