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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.polygamma"></a><a class="link" href="polygamma.html" title="Polygamma">Polygamma</a> | |
28 | </h3></div></div></div> | |
29 | <h5> | |
30 | <a name="math_toolkit.sf_gamma.polygamma.h0"></a> | |
31 | <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.synopsis"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.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">polygamma</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">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</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">polygamma</span><span class="special">(</span><span class="keyword">int</span> <span class="identifier">n</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.polygamma.h1"></a> | |
47 | <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.description"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.description">Description</a> | |
48 | </h5> | |
49 | <p> | |
50 | Returns the polygamma function of <span class="emphasis"><em>x</em></span>. Polygamma is defined | |
51 | as the n'th derivative of the digamma function: | |
52 | </p> | |
53 | <p> | |
54 | <span class="inlinemediaobject"><img src="../../../equations/polygamma1.svg"></span> | |
55 | </p> | |
56 | <p> | |
57 | The following graphs illustrate the behaviour of the function for odd and | |
58 | even order: | |
59 | </p> | |
60 | <p> | |
61 | <span class="inlinemediaobject"><img src="../../../graphs/polygamma2.svg" align="middle"></span> | |
62 | <span class="inlinemediaobject"><img src="../../../graphs/polygamma3.svg" align="middle"></span> | |
63 | </p> | |
64 | <p> | |
65 | The final <a class="link" href="../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can | |
66 | be used to control the behaviour of the function: how it handles errors, | |
67 | 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 | |
68 | documentation for more details</a>. | |
69 | </p> | |
70 | <p> | |
71 | 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 | |
72 | 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 | |
73 | T otherwise. | |
74 | </p> | |
75 | <h5> | |
76 | <a name="math_toolkit.sf_gamma.polygamma.h2"></a> | |
77 | <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.accuracy"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.accuracy">Accuracy</a> | |
78 | </h5> | |
79 | <p> | |
80 | The following table shows the peak errors (in units of epsilon) found on | |
81 | various platforms with various floating point types. Unless otherwise specified | |
82 | 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>. | |
83 | </p> | |
84 | <div class="table"> | |
85 | <a name="math_toolkit.sf_gamma.polygamma.table_polygamma"></a><p class="title"><b>Table 6.6. Error rates for polygamma</b></p> | |
86 | <div class="table-contents"><table class="table" summary="Error rates for polygamma"> | |
87 | <colgroup> | |
88 | <col> | |
89 | <col> | |
90 | <col> | |
91 | <col> | |
92 | <col> | |
93 | </colgroup> | |
94 | <thead><tr> | |
95 | <th> | |
96 | </th> | |
97 | <th> | |
98 | <p> | |
99 | Microsoft Visual C++ version 12.0<br> Win32<br> double | |
100 | </p> | |
101 | </th> | |
102 | <th> | |
103 | <p> | |
104 | GNU C++ version 5.1.0<br> linux<br> double | |
105 | </p> | |
106 | </th> | |
107 | <th> | |
108 | <p> | |
109 | GNU C++ version 5.1.0<br> linux<br> long double | |
110 | </p> | |
111 | </th> | |
112 | <th> | |
113 | <p> | |
114 | Sun compiler version 0x5130<br> Sun Solaris<br> long double | |
115 | </p> | |
116 | </th> | |
117 | </tr></thead> | |
118 | <tbody> | |
119 | <tr> | |
120 | <td> | |
121 | <p> | |
122 | Mathematica Data | |
123 | </p> | |
124 | </td> | |
125 | <td> | |
126 | <p> | |
127 | <span class="blue">Max = 6.34ε (Mean = 1.53ε)</span> | |
128 | </p> | |
129 | </td> | |
130 | <td> | |
131 | <p> | |
132 | <span class="blue">Max = 0.824ε (Mean = 0.0574ε)</span><br> | |
133 | <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> Max = 108ε (Mean = 15.2ε))<br> | |
134 | (<span class="emphasis"><em>GSL 1.16:</em></span> Max = 62.9ε (Mean = 12.8ε)) | |
135 | </p> | |
136 | </td> | |
137 | <td> | |
138 | <p> | |
139 | <span class="blue">Max = 7.38ε (Mean = 1.84ε)</span> | |
140 | </p> | |
141 | </td> | |
142 | <td> | |
143 | <p> | |
144 | <span class="blue">Max = 18.3ε (Mean = 4.16ε)</span> | |
145 | </p> | |
146 | </td> | |
147 | </tr> | |
148 | <tr> | |
149 | <td> | |
150 | <p> | |
151 | Mathematica Data - large arguments | |
152 | </p> | |
153 | </td> | |
154 | <td> | |
155 | <p> | |
156 | <span class="blue">Max = 150ε (Mean = 15.1ε)</span> | |
157 | </p> | |
158 | </td> | |
159 | <td> | |
160 | <p> | |
161 | <span class="blue">Max = 0.998ε (Mean = 0.0592ε)</span><br> | |
162 | <br> (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max | |
163 | = 1.71e+56ε (Mean = 1.01e+55ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_large_arguments">And | |
164 | other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span> | |
165 | Max = 244ε (Mean = 32.8ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_large_arguments">And | |
166 | other failures.</a>) | |
167 | </p> | |
168 | </td> | |
169 | <td> | |
170 | <p> | |
171 | <span class="blue">Max = 2.23ε (Mean = 0.323ε)</span> | |
172 | </p> | |
173 | </td> | |
174 | <td> | |
175 | <p> | |
176 | <span class="blue">Max = 2.35ε (Mean = 0.34ε)</span> | |
177 | </p> | |
178 | </td> | |
179 | </tr> | |
180 | <tr> | |
181 | <td> | |
182 | <p> | |
183 | Mathematica Data - negative arguments | |
184 | </p> | |
185 | </td> | |
186 | <td> | |
187 | <p> | |
188 | <span class="blue">Max = 497ε (Mean = 129ε)</span> | |
189 | </p> | |
190 | </td> | |
191 | <td> | |
192 | <p> | |
193 | <span class="blue">Max = 0.516ε (Mean = 0.022ε)</span><br> <br> | |
194 | (<span class="emphasis"><em>Rmath 3.0.2:</em></span> <span class="red">Max = +INFε (Mean | |
195 | = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_negative_arguments">And | |
196 | other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span> | |
197 | Max = 36.6ε (Mean = 3.04ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_negative_arguments">And | |
198 | other failures.</a>) | |
199 | </p> | |
200 | </td> | |
201 | <td> | |
202 | <p> | |
203 | <span class="blue">Max = 269ε (Mean = 87.7ε)</span> | |
204 | </p> | |
205 | </td> | |
206 | <td> | |
207 | <p> | |
208 | <span class="blue">Max = 269ε (Mean = 87.9ε)</span> | |
209 | </p> | |
210 | </td> | |
211 | </tr> | |
212 | <tr> | |
213 | <td> | |
214 | <p> | |
215 | Mathematica Data - large negative arguments | |
216 | </p> | |
217 | </td> | |
218 | <td> | |
219 | <p> | |
220 | <span class="blue">Max = 162ε (Mean = 101ε)</span> | |
221 | </p> | |
222 | </td> | |
223 | <td> | |
224 | <p> | |
225 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath | |
226 | 3.0.2:</em></span> <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_large_negative_arguments">And | |
227 | other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span> | |
228 | Max = 1.79ε (Mean = 0.197ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_large_negative_arguments">And | |
229 | other failures.</a>) | |
230 | </p> | |
231 | </td> | |
232 | <td> | |
233 | <p> | |
234 | <span class="blue">Max = 155ε (Mean = 96.4ε)</span> | |
235 | </p> | |
236 | </td> | |
237 | <td> | |
238 | <p> | |
239 | <span class="blue">Max = 155ε (Mean = 96.4ε)</span> | |
240 | </p> | |
241 | </td> | |
242 | </tr> | |
243 | <tr> | |
244 | <td> | |
245 | <p> | |
246 | Mathematica Data - small arguments | |
247 | </p> | |
248 | </td> | |
249 | <td> | |
250 | <p> | |
251 | <span class="blue">Max = 3ε (Mean = 0.496ε)</span> | |
252 | </p> | |
253 | </td> | |
254 | <td> | |
255 | <p> | |
256 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath | |
257 | 3.0.2:</em></span> Max = 106ε (Mean = 20ε))<br> (<span class="emphasis"><em>GSL 1.16:</em></span> | |
258 | Max = 15.2ε (Mean = 5.03ε)) | |
259 | </p> | |
260 | </td> | |
261 | <td> | |
262 | <p> | |
263 | <span class="blue">Max = 3.33ε (Mean = 0.75ε)</span> | |
264 | </p> | |
265 | </td> | |
266 | <td> | |
267 | <p> | |
268 | <span class="blue">Max = 3.33ε (Mean = 0.75ε)</span> | |
269 | </p> | |
270 | </td> | |
271 | </tr> | |
272 | <tr> | |
273 | <td> | |
274 | <p> | |
275 | Mathematica Data - Large orders and other bug cases | |
276 | </p> | |
277 | </td> | |
278 | <td> | |
279 | <p> | |
280 | <span class="blue">Max = 200ε (Mean = 57.2ε)</span> | |
281 | </p> | |
282 | </td> | |
283 | <td> | |
284 | <p> | |
285 | <span class="blue">Max = 0ε (Mean = 0ε)</span><br> <br> (<span class="emphasis"><em>Rmath | |
286 | 3.0.2:</em></span> <span class="red">Max = +INFε (Mean = +INFε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_Rmath_3_0_2_Mathematica_Data_Large_orders_and_other_bug_cases">And | |
287 | other failures.</a>)</span><br> (<span class="emphasis"><em>GSL 1.16:</em></span> | |
288 | Max = 151ε (Mean = 39.3ε) <a class="link" href="../logs_and_tables/logs.html#errors_GNU_C_version_5_1_0_linux_double_polygamma_GSL_1_16_Mathematica_Data_Large_orders_and_other_bug_cases">And | |
289 | other failures.</a>) | |
290 | </p> | |
291 | </td> | |
292 | <td> | |
293 | <p> | |
294 | <span class="blue">Max = 54.5ε (Mean = 13.3ε)</span> | |
295 | </p> | |
296 | </td> | |
297 | <td> | |
298 | <p> | |
299 | <span class="blue">Max = 90.1ε (Mean = 30.6ε)</span> | |
300 | </p> | |
301 | </td> | |
302 | </tr> | |
303 | </tbody> | |
304 | </table></div> | |
305 | </div> | |
306 | <br class="table-break"><p> | |
307 | As shown above, error rates are generally very acceptable for moderately | |
308 | sized arguments. Error rates should stay low for exact inputs, however, please | |
309 | note that the function becomes exceptionally sensitive to small changes in | |
310 | input for large n and negative x, indeed for cases where <span class="emphasis"><em>n!</em></span> | |
311 | would overflow, the function changes directly from -∞ to +∞ somewhere between | |
312 | each negative integer - <span class="emphasis"><em>these cases are not handled correctly</em></span>. | |
313 | </p> | |
314 | <p> | |
315 | <span class="bold"><strong>For these reasons results should be treated with extreme | |
316 | caution when <span class="emphasis"><em>n</em></span> is large and x negative</strong></span>. | |
317 | </p> | |
318 | <h5> | |
319 | <a name="math_toolkit.sf_gamma.polygamma.h3"></a> | |
320 | <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.testing"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.testing">Testing</a> | |
321 | </h5> | |
322 | <p> | |
323 | Testing is against Mathematica generated spot values to 35 digit precision. | |
324 | </p> | |
325 | <h5> | |
326 | <a name="math_toolkit.sf_gamma.polygamma.h4"></a> | |
327 | <span class="phrase"><a name="math_toolkit.sf_gamma.polygamma.implementation"></a></span><a class="link" href="polygamma.html#math_toolkit.sf_gamma.polygamma.implementation">Implementation</a> | |
328 | </h5> | |
329 | <p> | |
330 | For x < 0 the following reflection formula is used: | |
331 | </p> | |
332 | <p> | |
333 | <span class="inlinemediaobject"><img src="../../../equations/polygamma2.svg"></span> | |
334 | </p> | |
335 | <p> | |
336 | The n'th derivative of <span class="emphasis"><em>cot(x)</em></span> is tabulated for small | |
337 | <span class="emphasis"><em>n</em></span>, and for larger n has the general form: | |
338 | </p> | |
339 | <p> | |
340 | <span class="inlinemediaobject"><img src="../../../equations/polygamma3.svg"></span> | |
341 | </p> | |
342 | <p> | |
343 | The coefficients of the cosine terms can be calculated iteratively starting | |
344 | from <span class="emphasis"><em>C<sub>1,0</sub> = -1</em></span> and then using | |
345 | </p> | |
346 | <p> | |
347 | <span class="inlinemediaobject"><img src="../../../equations/polygamma7.svg"></span> | |
348 | </p> | |
349 | <p> | |
350 | to generate coefficients for n+1. | |
351 | </p> | |
352 | <p> | |
353 | Note that every other coefficient is zero, and therefore what we have are | |
354 | even or odd polynomials depending on whether n is even or odd. | |
355 | </p> | |
356 | <p> | |
357 | Once x is positive then we have two methods available to us, for small x | |
358 | we use the series expansion: | |
359 | </p> | |
360 | <p> | |
361 | <span class="inlinemediaobject"><img src="../../../equations/polygamma4.svg"></span> | |
362 | </p> | |
363 | <p> | |
364 | Note that the evaluation of zeta functions at integer values is essentially | |
365 | a table lookup as <a class="link" href="../zetas/zeta.html" title="Riemann Zeta Function">zeta</a> is | |
366 | optimized for those cases. | |
367 | </p> | |
368 | <p> | |
369 | For large x we use the asymptotic expansion: | |
370 | </p> | |
371 | <p> | |
372 | <span class="inlinemediaobject"><img src="../../../equations/polygamma5.svg"></span> | |
373 | </p> | |
374 | <p> | |
375 | For x in-between the two extremes we use the relation: | |
376 | </p> | |
377 | <p> | |
378 | <span class="inlinemediaobject"><img src="../../../equations/polygamma6.svg"></span> | |
379 | </p> | |
380 | <p> | |
381 | to make x large enough for the asymptotic expansion to be used. | |
382 | </p> | |
383 | <p> | |
384 | There are also two special cases: | |
385 | </p> | |
386 | <p> | |
387 | <span class="inlinemediaobject"><img src="../../../equations/polygamma8.svg"></span> | |
388 | </p> | |
389 | <p> | |
390 | <span class="inlinemediaobject"><img src="../../../equations/polygamma9.svg"></span> | |
391 | </p> | |
392 | </div> | |
393 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
394 | <td align="left"></td> | |
395 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
396 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
397 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
398 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
399 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
400 | 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>) | |
401 | </p> | |
402 | </div></td> | |
403 | </tr></table> | |
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405 | <div class="spirit-nav"> | |
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