libs/math/doc/html/math_toolkit/sf_gamma/trigamma.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   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 2.11.0">
   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">&lt;</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">&gt;</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">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</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">&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;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</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&#160;20.&#160;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&#160;20.&#160;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&#160;8.5.&#160;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 = 0.998&#949; (Mean = 0.105&#949;)</span><br> <br>
 124                   (<span class="emphasis"><em>GSL 2.1:</em></span> Max = 1.34e+04&#949; (Mean = 1.49e+03&#949;))<br>
 125                   (<span class="emphasis"><em>Rmath 3.2.3:</em></span> Max = 1.34e+04&#949; (Mean = 1.51e+03&#949;))
 126                 </p>
 127               </td>
 128 <td>
 129                 <p>
 130                   <span class="blue">Max = 1.28&#949; (Mean = 0.449&#949;)</span>
 131                 </p>
 132               </td>
 133 <td>
 134                 <p>
 135                   <span class="blue">Max = 1.28&#949; (Mean = 0.449&#949;)</span>
 136                 </p>
 137               </td>
 138 <td>
 139                 <p>
 140                   <span class="blue">Max = 1&#949; (Mean = 0.382&#949;)</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 &gt;= 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 &#169; 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       R&#229;de, 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>