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4 | <title>Trigamma</title> | |
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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">z</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 15. 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">z</span><span class="special">,</span> <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 15. 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 | <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 15. 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 15. 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 6.5. 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 = 1ε (Mean = 0.382ε)</span> | |
122 | </p> | |
123 | </td> | |
124 | <td> | |
125 | <p> | |
126 | <span class="blue">Max = 0.998ε (Mean = 0.105ε)</span><br> <br> | |
127 | (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 1.34e+04ε (Mean = 1.51e+03ε))<br> | |
128 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 1.34e+04ε (Mean = 1.49e+03ε)) | |
129 | </p> | |
130 | </td> | |
131 | <td> | |
132 | <p> | |
133 | <span class="blue">Max = 1.28ε (Mean = 0.449ε)</span> | |
134 | </p> | |
135 | </td> | |
136 | <td> | |
137 | <p> | |
138 | <span class="blue">Max = 1.28ε (Mean = 0.447ε)</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 >= 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 © 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 Råde, 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> | |
199 | </tr></table> | |
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