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| <div class="titlepage"><div><div><h3 class="title"> |
| <a name="math_toolkit.sf_gamma.trigamma"></a><a class="link" href="trigamma.html" title="Trigamma">Trigamma</a> |
| </h3></div></div></div> |
| <h5> |
| <a name="math_toolkit.sf_gamma.trigamma.h0"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.synopsis"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.synopsis">Synopsis</a> |
| </h5> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">special_functions</span><span class="special">/</span><span class="identifier">trigamma</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| </pre> |
| <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">);</span> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> |
| <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>calculated-result-type</em></span></a> <span class="identifier">trigamma</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces</span> |
| </pre> |
| <h5> |
| <a name="math_toolkit.sf_gamma.trigamma.h1"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.description"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.description">Description</a> |
| </h5> |
| <p> |
| Returns the trigamma function of <span class="emphasis"><em>x</em></span>. Trigamma is defined |
| as the derivative of the digamma function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/trigamma1.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 14. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can |
| be used to control the behaviour of the function: how it handles errors, |
| what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 14. Policies: Controlling Precision, Error Handling etc">policy |
| documentation for more details</a>. |
| </p> |
| <p> |
| The return type of this function is computed using the <a class="link" href="../result_type.html" title="Calculation of the Type of the Result"><span class="emphasis"><em>result |
| type calculation rules</em></span></a>: the result is of type <code class="computeroutput"><span class="keyword">double</span></code> when T is an integer type, and type |
| T otherwise. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_gamma.trigamma.h2"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.accuracy"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of epsilon) found on |
| various platforms with various floating point types. Unless otherwise specified |
| any floating point type that is narrower than the one shown will have <a class="link" href="../relative_error.html#math_toolkit.relative_error.zero_error">effectively zero error</a>. |
| </p> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| Random Values |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32 Visual C++ 12 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.0 Mean=0.4 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win64 Mingw GCC |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.4 Mean=0.4 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win64 Mingw GCC __float128 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.0 Mean=0.5 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <p> |
| As shown above, error rates are generally very low for built in types. For |
| multiprecision types, error rates are typically in the order of a few epsilon. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_gamma.trigamma.h3"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.testing"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.testing">Testing</a> |
| </h5> |
| <p> |
| Testing is against Mathematica generated spot values to 35 digit precision. |
| </p> |
| <h5> |
| <a name="math_toolkit.sf_gamma.trigamma.h4"></a> |
| <span class="phrase"><a name="math_toolkit.sf_gamma.trigamma.implementation"></a></span><a class="link" href="trigamma.html#math_toolkit.sf_gamma.trigamma.implementation">Implementation</a> |
| </h5> |
| <p> |
| The 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 |
| via: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/trigamma2.svg"></span> |
| </p> |
| <p> |
| Then arguments in the range [0, 1) are shifted to >= 1 via: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/trigamma3.svg"></span> |
| </p> |
| <p> |
| Then evaluation is via one of a number of rational approximations, for small |
| x these are of the form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/trigamma4.svg"></span> |
| </p> |
| <p> |
| and for large x of the form: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../equations/trigamma5.svg"></span> |
| </p> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) |
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