1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Trigamma</title> 5<link rel="stylesheet" href="../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../sf_gamma.html" title="Gamma Functions"> 9<link rel="prev" href="digamma.html" title="Digamma"> 10<link rel="next" href="polygamma.html" title="Polygamma"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="polygamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="math_toolkit.sf_gamma.trigamma"></a><a class="link" href="trigamma.html" title="Trigamma">Trigamma</a> 28</h3></div></div></div> 29<h5> 30<a name="math_toolkit.sf_gamma.trigamma.h0"></a> 31 <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.synopsis"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.synopsis">Synopsis</a> 32 </h5> 33<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">trigamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> 34</pre> 35<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> 36 37<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> 38<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> 39 40<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 41<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">&);</span> 42 43<span class="special">}}</span> <span class="comment">// namespaces</span> 44</pre> 45<h5> 46<a name="math_toolkit.sf_gamma.trigamma.h1"></a> 47 <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.description"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.description">Description</a> 48 </h5> 49<p> 50 Returns the trigamma function of <span class="emphasis"><em>x</em></span>. Trigamma is defined 51 as the derivative of the digamma function: 52 </p> 53<div class="blockquote"><blockquote class="blockquote"><p> 54 <span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span> 55 56 </p></blockquote></div> 57<div class="blockquote"><blockquote class="blockquote"><p> 58 <span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span> 59 60 </p></blockquote></div> 61<p> 62 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 63 be used to control the behaviour of the function: how it handles errors, 64 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 65 documentation for more details</a>. 66 </p> 67<p> 68 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 69 type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type 70 T otherwise. 71 </p> 72<h5> 73<a name="math_toolkit.sf_gamma.trigamma.h2"></a> 74 <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.accuracy"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.accuracy">Accuracy</a> 75 </h5> 76<p> 77 The following table shows the peak errors (in units of epsilon) found on 78 various platforms with various floating point types. Unless otherwise specified 79 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>. 80 </p> 81<div class="table"> 82<a name="math_toolkit.sf_gamma.trigamma.table_trigamma"></a><p class="title"><b>Table 8.5. Error rates for trigamma</b></p> 83<div class="table-contents"><table class="table" summary="Error rates for trigamma"> 84<colgroup> 85<col> 86<col> 87<col> 88<col> 89<col> 90</colgroup> 91<thead><tr> 92<th> 93 </th> 94<th> 95 <p> 96 GNU C++ version 7.1.0<br> linux<br> double 97 </p> 98 </th> 99<th> 100 <p> 101 GNU C++ version 7.1.0<br> linux<br> long double 102 </p> 103 </th> 104<th> 105 <p> 106 Sun compiler version 0x5150<br> Sun Solaris<br> long double 107 </p> 108 </th> 109<th> 110 <p> 111 Microsoft Visual C++ version 14.1<br> Win32<br> double 112 </p> 113 </th> 114</tr></thead> 115<tbody><tr> 116<td> 117 <p> 118 Mathematica Data 119 </p> 120 </td> 121<td> 122 <p> 123 <span class="blue">Max = 0.998ε (Mean = 0.105ε)</span><br> <br> 124 (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.34e+04ε (Mean = 1.49e+03ε))<br> 125 (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.34e+04ε (Mean = 1.51e+03ε)) 126 </p> 127 </td> 128<td> 129 <p> 130 <span class="blue">Max = 1.28ε (Mean = 0.449ε)</span> 131 </p> 132 </td> 133<td> 134 <p> 135 <span class="blue">Max = 1.28ε (Mean = 0.449ε)</span> 136 </p> 137 </td> 138<td> 139 <p> 140 <span class="blue">Max = 1ε (Mean = 0.382ε)</span> 141 </p> 142 </td> 143</tr></tbody> 144</table></div> 145</div> 146<br class="table-break"><p> 147 As shown above, error rates are generally very low for built in types. For 148 multiprecision types, error rates are typically in the order of a few epsilon. 149 </p> 150<p> 151 The following error plot are based on an exhaustive search of the functions 152 domain, MSVC-15.5 at <code class="computeroutput"><span class="keyword">double</span></code> 153 precision, and GCC-7.1/Ubuntu for <code class="computeroutput"><span class="keyword">long</span> 154 <span class="keyword">double</span></code> and <code class="computeroutput"><span class="identifier">__float128</span></code>. 155 </p> 156<div class="blockquote"><blockquote class="blockquote"><p> 157 <span class="inlinemediaobject"><img src="../../../graphs/trigamma__double.svg" align="middle"></span> 158 159 </p></blockquote></div> 160<div class="blockquote"><blockquote class="blockquote"><p> 161 <span class="inlinemediaobject"><img src="../../../graphs/trigamma__80_bit_long_double.svg" align="middle"></span> 162 163 </p></blockquote></div> 164<div class="blockquote"><blockquote class="blockquote"><p> 165 <span class="inlinemediaobject"><img src="../../../graphs/trigamma____float128.svg" align="middle"></span> 166 167 </p></blockquote></div> 168<h5> 169<a name="math_toolkit.sf_gamma.trigamma.h3"></a> 170 <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> 171 </h5> 172<p> 173 Testing is against Mathematica generated spot values to 35 digit precision. 174 </p> 175<h5> 176<a name="math_toolkit.sf_gamma.trigamma.h4"></a> 177 <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.implementation"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.implementation">Implementation</a> 178 </h5> 179<p> 180 The arbitrary precision version of this function simply calls <a class="link" href="polygamma.html" title="Polygamma">polygamma</a>. 181 </p> 182<p> 183 For built in fixed-precision types, negative arguments are first made positive 184 via: 185 </p> 186<div class="blockquote"><blockquote class="blockquote"><p> 187 <span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span> 188 189 </p></blockquote></div> 190<p> 191 Then arguments in the range [0, 1) are shifted to >= 1 via: 192 </p> 193<div class="blockquote"><blockquote class="blockquote"><p> 194 <span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span> 195 196 </p></blockquote></div> 197<p> 198 Then evaluation is via one of a number of rational approximations, for small 199 <span class="emphasis"><em>x</em></span> these are of the form: 200 </p> 201<div class="blockquote"><blockquote class="blockquote"><p> 202 <span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span> 203 204 </p></blockquote></div> 205<p> 206 and for large <span class="emphasis"><em>x</em></span> of the form: 207 </p> 208<div class="blockquote"><blockquote class="blockquote"><p> 209 <span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span> 210 211 </p></blockquote></div> 212</div> 213<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 214<td align="left"></td> 215<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 216 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 217 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 218 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 219 Daryle Walker and Xiaogang Zhang<p> 220 Distributed under the Boost Software License, Version 1.0. (See accompanying 221 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>) 222 </p> 223</div></td> 224</tr></table> 225<hr> 226<div class="spirit-nav"> 227<a accesskey="p" href="digamma.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../sf_gamma.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="polygamma.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 228</div> 229</body> 230</html> 231
