ceph/src/boost/libs/math/doc/html/math_toolkit/sf_gamma/trigamma.html

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  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">z</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;15.&#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">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;15.&#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 <p>
  54         <span class="inlinemediaobject"><img src="../../../equations/trigamma1.svg"></span>
  55       </p>
  56 <p>
  57         <span class="inlinemediaobject"><img src="../../../graphs/trigamma.svg" align="middle"></span>
  58       </p>
  59 <p>
  60         The final <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
  61         be used to control the behaviour of the function: how it handles errors,
  62         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;15.&#160;Policies: Controlling Precision, Error Handling etc">policy
  63         documentation for more details</a>.
  64       </p>
  65 <p>
  66         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
  67         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
  68         T otherwise.
  69       </p>
  70 <h5>
  71 <a name="math_toolkit.sf_gamma.trigamma.h2"></a>
  72         <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>
  73       </h5>
  74 <p>
  75         The following table shows the peak errors (in units of epsilon) found on
  76         various platforms with various floating point types. Unless otherwise specified
  77         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>.
  78       </p>
  79 <div class="table">
  80 <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>
  81 <div class="table-contents"><table class="table" summary="Error rates for trigamma">
  82 <colgroup>
  83 <col>
  84 <col>
  85 <col>
  86 <col>
  87 <col>
  88 </colgroup>
  89 <thead><tr>
  90 <th>
  91               </th>
  92 <th>
  93                 <p>
  94                   Microsoft Visual C++ version 12.0<br> Win32<br> double
  95                 </p>
  96               </th>
  97 <th>
  98                 <p>
  99                   GNU C++ version 5.1.0<br> linux<br> double
 100                 </p>
 101               </th>
 102 <th>
 103                 <p>
 104                   GNU C++ version 5.1.0<br> linux<br> long double
 105                 </p>
 106               </th>
 107 <th>
 108                 <p>
 109                   Sun compiler version 0x5130<br> Sun Solaris<br> long double
 110                 </p>
 111               </th>
 112 </tr></thead>
 113 <tbody><tr>
 114 <td>
 115                 <p>
 116                   Mathematica Data
 117                 </p>
 118               </td>
 119 <td>
 120                 <p>
 121                   <span class="blue">Max = 1&#949; (Mean = 0.382&#949;)</span>
 122                 </p>
 123               </td>
 124 <td>
 125                 <p>
 126                   <span class="blue">Max = 0.998&#949; (Mean = 0.105&#949;)</span><br> <br>
 127                   (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.34e+04&#949; (Mean = 1.51e+03&#949;))<br>
 128                   (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.34e+04&#949; (Mean = 1.49e+03&#949;))
 129                 </p>
 130               </td>
 131 <td>
 132                 <p>
 133                   <span class="blue">Max = 1.28&#949; (Mean = 0.449&#949;)</span>
 134                 </p>
 135               </td>
 136 <td>
 137                 <p>
 138                   <span class="blue">Max = 1.28&#949; (Mean = 0.447&#949;)</span>
 139                 </p>
 140               </td>
 141 </tr></tbody>
 142 </table></div>
 143 </div>
 144 <br class="table-break"><p>
 145         As shown above, error rates are generally very low for built in types. For
 146         multiprecision types, error rates are typically in the order of a few epsilon.
 147       </p>
 148 <h5>
 149 <a name="math_toolkit.sf_gamma.trigamma.h3"></a>
 150         <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>
 151       </h5>
 152 <p>
 153         Testing is against Mathematica generated spot values to 35 digit precision.
 154       </p>
 155 <h5>
 156 <a name="math_toolkit.sf_gamma.trigamma.h4"></a>
 157         <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>
 158       </h5>
 159 <p>
 160         The arbitrary precision version of this function simply calls <a class="link" href="polygamma.html" title="Polygamma">polygamma</a>.
 161       </p>
 162 <p>
 163         For built in fixed precision types, negative arguments are first made positive
 164         via:
 165       </p>
 166 <p>
 167         <span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span>
 168       </p>
 169 <p>
 170         Then arguments in the range [0, 1) are shifted to &gt;= 1 via:
 171       </p>
 172 <p>
 173         <span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span>
 174       </p>
 175 <p>
 176         Then evaluation is via one of a number of rational approximations, for small
 177         x these are of the form:
 178       </p>
 179 <p>
 180         <span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span>
 181       </p>
 182 <p>
 183         and for large x of the form:
 184       </p>
 185 <p>
 186         <span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span>
 187       </p>
 188 </div>
 189 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 190 <td align="left"></td>
 191 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 192       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 193       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 194       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 195         Distributed under the Boost Software License, Version 1.0. (See accompanying
 196         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>)
 197       </p>
 198 </div></td>
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