1// (C) Copyright John Maddock 2006.
2// Use, modification and distribution are subject to the
3// Boost Software License, Version 1.0. (See accompanying file
4// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
5
6#ifndef BOOST_MATH_SF_TRIGAMMA_HPP
7#define BOOST_MATH_SF_TRIGAMMA_HPP
8
9#ifdef _MSC_VER
10#pragma once
11#endif
12
13#include <boost/math/special_functions/math_fwd.hpp>
14#include <boost/math/tools/rational.hpp>
15#include <boost/math/tools/series.hpp>
16#include <boost/math/tools/promotion.hpp>
17#include <boost/math/policies/error_handling.hpp>
18#include <boost/math/constants/constants.hpp>
19#include <boost/mpl/comparison.hpp>
20#include <boost/math/tools/big_constant.hpp>
21#include <boost/math/special_functions/polygamma.hpp>
22
23#if defined(__GNUC__) && defined(BOOST_MATH_USE_FLOAT128)
24//
25// This is the only way we can avoid
26// warning: non-standard suffix on floating constant [-Wpedantic]
27// when building with -Wall -pedantic. Neither __extension__
28// nor #pragma diagnostic ignored work :(
29//
30#pragma GCC system_header
31#endif
32
33namespace boost{
34namespace math{
35namespace detail{
36
37template<class T, class Policy>
38T polygamma_imp(const int n, T x, const Policy &pol);
39
40template <class T, class Policy>
41T trigamma_prec(T x, const boost::integral_constant<int, 53>*, const Policy&)
42{
43 // Max error in interpolated form: 3.736e-017
44 static const T offset = BOOST_MATH_BIG_CONSTANT(T, 53, 2.1093254089355469);
45 static const T P_1_2[] = {
46 BOOST_MATH_BIG_CONSTANT(T, 53, -1.1093280605946045),
47 BOOST_MATH_BIG_CONSTANT(T, 53, -3.8310674472619321),
48 BOOST_MATH_BIG_CONSTANT(T, 53, -3.3703848401898283),
49 BOOST_MATH_BIG_CONSTANT(T, 53, 0.28080574467981213),
50 BOOST_MATH_BIG_CONSTANT(T, 53, 1.6638069578676164),
51 BOOST_MATH_BIG_CONSTANT(T, 53, 0.64468386819102836),
52 };
53 static const T Q_1_2[] = {
54 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
55 BOOST_MATH_BIG_CONSTANT(T, 53, 3.4535389668541151),
56 BOOST_MATH_BIG_CONSTANT(T, 53, 4.5208926987851437),
57 BOOST_MATH_BIG_CONSTANT(T, 53, 2.7012734178351534),
58 BOOST_MATH_BIG_CONSTANT(T, 53, 0.64468798399785611),
59 BOOST_MATH_BIG_CONSTANT(T, 53, -0.20314516859987728e-6),
60 };
61 // Max error in interpolated form: 1.159e-017
62 static const T P_2_4[] = {
63 BOOST_MATH_BIG_CONSTANT(T, 53, -0.13803835004508849e-7),
64 BOOST_MATH_BIG_CONSTANT(T, 53, 0.50000049158540261),
65 BOOST_MATH_BIG_CONSTANT(T, 53, 1.6077979838469348),
66 BOOST_MATH_BIG_CONSTANT(T, 53, 2.5645435828098254),
67 BOOST_MATH_BIG_CONSTANT(T, 53, 2.0534873203680393),
68 BOOST_MATH_BIG_CONSTANT(T, 53, 0.74566981111565923),
69 };
70 static const T Q_2_4[] = {
71 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
72 BOOST_MATH_BIG_CONSTANT(T, 53, 2.8822787662376169),
73 BOOST_MATH_BIG_CONSTANT(T, 53, 4.1681660554090917),
74 BOOST_MATH_BIG_CONSTANT(T, 53, 2.7853527819234466),
75 BOOST_MATH_BIG_CONSTANT(T, 53, 0.74967671848044792),
76 BOOST_MATH_BIG_CONSTANT(T, 53, -0.00057069112416246805),
77 };
78 // Maximum Deviation Found: 6.896e-018
79 // Expected Error Term : -6.895e-018
80 // Maximum Relative Change in Control Points : 8.497e-004
81 static const T P_4_inf[] = {
82 static_cast<T>(0.68947581948701249e-17L),
83 static_cast<T>(0.49999999999998975L),
84 static_cast<T>(1.0177274392923795L),
85 static_cast<T>(2.498208511343429L),
86 static_cast<T>(2.1921221359427595L),
87 static_cast<T>(1.5897035272532764L),
88 static_cast<T>(0.40154388356961734L),
89 };
90 static const T Q_4_inf[] = {
91 static_cast<T>(1.0L),
92 static_cast<T>(1.7021215452463932L),
93 static_cast<T>(4.4290431747556469L),
94 static_cast<T>(2.9745631894384922L),
95 static_cast<T>(2.3013614809773616L),
96 static_cast<T>(0.28360399799075752L),
97 static_cast<T>(0.022892987908906897L),
98 };
99
100 if(x <= 2)
101 {
102 return (offset + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x);
103 }
104 else if(x <= 4)
105 {
106 T y = 1 / x;
107 return (1 + tools::evaluate_polynomial(P_2_4, y) / tools::evaluate_polynomial(Q_2_4, y)) / x;
108 }
109 T y = 1 / x;
110 return (1 + tools::evaluate_polynomial(P_4_inf, y) / tools::evaluate_polynomial(Q_4_inf, y)) / x;
111}
112
113template <class T, class Policy>
114T trigamma_prec(T x, const boost::integral_constant<int, 64>*, const Policy&)
115{
116 // Max error in interpolated form: 1.178e-020
117 static const T offset_1_2 = BOOST_MATH_BIG_CONSTANT(T, 64, 2.109325408935546875);
118 static const T P_1_2[] = {
119 BOOST_MATH_BIG_CONSTANT(T, 64, -1.10932535608960258341),
120 BOOST_MATH_BIG_CONSTANT(T, 64, -4.18793841543017129052),
121 BOOST_MATH_BIG_CONSTANT(T, 64, -4.63865531898487734531),
122 BOOST_MATH_BIG_CONSTANT(T, 64, -0.919832884430500908047),
123 BOOST_MATH_BIG_CONSTANT(T, 64, 1.68074038333180423012),
124 BOOST_MATH_BIG_CONSTANT(T, 64, 1.21172611429185622377),
125 BOOST_MATH_BIG_CONSTANT(T, 64, 0.259635673503366427284),
126 };
127 static const T Q_1_2[] = {
128 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
129 BOOST_MATH_BIG_CONSTANT(T, 64, 3.77521119359546982995),
130 BOOST_MATH_BIG_CONSTANT(T, 64, 5.664338024578956321),
131 BOOST_MATH_BIG_CONSTANT(T, 64, 4.25995134879278028361),
132 BOOST_MATH_BIG_CONSTANT(T, 64, 1.62956638448940402182),
133 BOOST_MATH_BIG_CONSTANT(T, 64, 0.259635512844691089868),
134 BOOST_MATH_BIG_CONSTANT(T, 64, 0.629642219810618032207e-8),
135 };
136 // Max error in interpolated form: 3.912e-020
137 static const T P_2_8[] = {
138 BOOST_MATH_BIG_CONSTANT(T, 64, -0.387540035162952880976e-11),
139 BOOST_MATH_BIG_CONSTANT(T, 64, 0.500000000276430504),
140 BOOST_MATH_BIG_CONSTANT(T, 64, 3.21926880986360957306),
141 BOOST_MATH_BIG_CONSTANT(T, 64, 10.2550347708483445775),
142 BOOST_MATH_BIG_CONSTANT(T, 64, 18.9002075150709144043),
143 BOOST_MATH_BIG_CONSTANT(T, 64, 21.0357215832399705625),
144 BOOST_MATH_BIG_CONSTANT(T, 64, 13.4346512182925923978),
145 BOOST_MATH_BIG_CONSTANT(T, 64, 3.98656291026448279118),
146 };
147 static const T Q_2_8[] = {
148 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
149 BOOST_MATH_BIG_CONSTANT(T, 64, 6.10520430478613667724),
150 BOOST_MATH_BIG_CONSTANT(T, 64, 18.475001060603645512),
151 BOOST_MATH_BIG_CONSTANT(T, 64, 31.7087534567758405638),
152 BOOST_MATH_BIG_CONSTANT(T, 64, 31.908814523890465398),
153 BOOST_MATH_BIG_CONSTANT(T, 64, 17.4175479039227084798),
154 BOOST_MATH_BIG_CONSTANT(T, 64, 3.98749106958394941276),
155 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000115917322224411128566),
156 };
157 // Maximum Deviation Found: 2.635e-020
158 // Expected Error Term : 2.635e-020
159 // Maximum Relative Change in Control Points : 1.791e-003
160 static const T P_8_inf[] = {
161 BOOST_MATH_BIG_CONSTANT(T, 64, -0.263527875092466899848e-19),
162 BOOST_MATH_BIG_CONSTANT(T, 64, 0.500000000000000058145),
163 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0730121433777364138677),
164 BOOST_MATH_BIG_CONSTANT(T, 64, 1.94505878379957149534),
165 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0517092358874932620529),
166 BOOST_MATH_BIG_CONSTANT(T, 64, 1.07995383547483921121),
167 };
168 static const T Q_8_inf[] = {
169 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
170 BOOST_MATH_BIG_CONSTANT(T, 64, -0.187309046577818095504),
171 BOOST_MATH_BIG_CONSTANT(T, 64, 3.95255391645238842975),
172 BOOST_MATH_BIG_CONSTANT(T, 64, -1.14743283327078949087),
173 BOOST_MATH_BIG_CONSTANT(T, 64, 2.52989799376344914499),
174 BOOST_MATH_BIG_CONSTANT(T, 64, -0.627414303172402506396),
175 BOOST_MATH_BIG_CONSTANT(T, 64, 0.141554248216425512536),
176 };
177
178 if(x <= 2)
179 {
180 return (offset_1_2 + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x);
181 }
182 else if(x <= 8)
183 {
184 T y = 1 / x;
185 return (1 + tools::evaluate_polynomial(P_2_8, y) / tools::evaluate_polynomial(Q_2_8, y)) / x;
186 }
187 T y = 1 / x;
188 return (1 + tools::evaluate_polynomial(P_8_inf, y) / tools::evaluate_polynomial(Q_8_inf, y)) / x;
189}
190
191template <class T, class Policy>
192T trigamma_prec(T x, const boost::integral_constant<int, 113>*, const Policy&)
193{
194 // Max error in interpolated form: 1.916e-035
195
196 static const T P_1_2[] = {
197 BOOST_MATH_BIG_CONSTANT(T, 113, -0.999999999999999082554457936871832533),
198 BOOST_MATH_BIG_CONSTANT(T, 113, -4.71237311120865266379041700054847734),
199 BOOST_MATH_BIG_CONSTANT(T, 113, -7.94125711970499027763789342500817316),
200 BOOST_MATH_BIG_CONSTANT(T, 113, -5.74657746697664735258222071695644535),
201 BOOST_MATH_BIG_CONSTANT(T, 113, -0.404213349456398905981223965160595687),
202 BOOST_MATH_BIG_CONSTANT(T, 113, 2.47877781178642876561595890095758896),
203 BOOST_MATH_BIG_CONSTANT(T, 113, 2.07714151702455125992166949812126433),
204 BOOST_MATH_BIG_CONSTANT(T, 113, 0.858877899162360138844032265418028567),
205 BOOST_MATH_BIG_CONSTANT(T, 113, 0.20499222604410032375789018837922397),
206 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0272103140348194747360175268778415049),
207 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0015764849020876949848954081173520686),
208 };
209 static const T Q_1_2[] = {
210 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
211 BOOST_MATH_BIG_CONSTANT(T, 113, 4.71237311120863419878375031457715223),
212 BOOST_MATH_BIG_CONSTANT(T, 113, 9.58619118655339853449127952145877467),
213 BOOST_MATH_BIG_CONSTANT(T, 113, 11.0940067269829372437561421279054968),
214 BOOST_MATH_BIG_CONSTANT(T, 113, 8.09075424749327792073276309969037885),
215 BOOST_MATH_BIG_CONSTANT(T, 113, 3.87705890159891405185343806884451286),
216 BOOST_MATH_BIG_CONSTANT(T, 113, 1.22758678701914477836330837816976782),
217 BOOST_MATH_BIG_CONSTANT(T, 113, 0.249092040606385004109672077814668716),
218 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0295750413900655597027079600025569048),
219 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00157648490200498142247694709728858139),
220 BOOST_MATH_BIG_CONSTANT(T, 113, 0.161264050344059471721062360645432809e-14),
221 };
222
223 // Max error in interpolated form: 8.958e-035
224 static const T P_2_4[] = {
225 BOOST_MATH_BIG_CONSTANT(T, 113, -2.55843734739907925764326773972215085),
226 BOOST_MATH_BIG_CONSTANT(T, 113, -12.2830208240542011967952466273455887),
227 BOOST_MATH_BIG_CONSTANT(T, 113, -23.9195022162767993526575786066414403),
228 BOOST_MATH_BIG_CONSTANT(T, 113, -24.9256431504823483094158828285470862),
229 BOOST_MATH_BIG_CONSTANT(T, 113, -14.7979122765478779075108064826412285),
230 BOOST_MATH_BIG_CONSTANT(T, 113, -4.46654453928610666393276765059122272),
231 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0191439033405649675717082465687845002),
232 BOOST_MATH_BIG_CONSTANT(T, 113, 0.515412052554351265708917209749037352),
233 BOOST_MATH_BIG_CONSTANT(T, 113, 0.195378348786064304378247325360320038),
234 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0334761282624174313035014426794245393),
235 BOOST_MATH_BIG_CONSTANT(T, 113, 0.002373665205942206348500250056602687),
236 };
237 static const T Q_2_4[] = {
238 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
239 BOOST_MATH_BIG_CONSTANT(T, 113, 4.80098558454419907830670928248659245),
240 BOOST_MATH_BIG_CONSTANT(T, 113, 9.99220727843170133895059300223445265),
241 BOOST_MATH_BIG_CONSTANT(T, 113, 11.8896146167631330735386697123464976),
242 BOOST_MATH_BIG_CONSTANT(T, 113, 8.96613256683809091593793565879092581),
243 BOOST_MATH_BIG_CONSTANT(T, 113, 4.47254136149624110878909334574485751),
244 BOOST_MATH_BIG_CONSTANT(T, 113, 1.48600982028196527372434773913633152),
245 BOOST_MATH_BIG_CONSTANT(T, 113, 0.319570735766764237068541501137990078),
246 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0407358345787680953107374215319322066),
247 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00237366520593271641375755486420859837),
248 BOOST_MATH_BIG_CONSTANT(T, 113, 0.239554887903526152679337256236302116e-15),
249 BOOST_MATH_BIG_CONSTANT(T, 113, -0.294749244740618656265237072002026314e-17),
250 };
251
252 static const T y_offset_2_4 = BOOST_MATH_BIG_CONSTANT(T, 113, 3.558437347412109375);
253
254 // Max error in interpolated form: 4.319e-035
255 static const T P_4_8[] = {
256 BOOST_MATH_BIG_CONSTANT(T, 113, 0.166626112697021464248967707021688845e-16),
257 BOOST_MATH_BIG_CONSTANT(T, 113, 0.499999999999997739552090249208808197),
258 BOOST_MATH_BIG_CONSTANT(T, 113, 6.40270945019053817915772473771553187),
259 BOOST_MATH_BIG_CONSTANT(T, 113, 41.3833374155000608013677627389343329),
260 BOOST_MATH_BIG_CONSTANT(T, 113, 166.803341854562809335667241074035245),
261 BOOST_MATH_BIG_CONSTANT(T, 113, 453.39964786925369319960722793414521),
262 BOOST_MATH_BIG_CONSTANT(T, 113, 851.153712317697055375935433362983944),
263 BOOST_MATH_BIG_CONSTANT(T, 113, 1097.70657567285059133109286478004458),
264 BOOST_MATH_BIG_CONSTANT(T, 113, 938.431232478455316020076349367632922),
265 BOOST_MATH_BIG_CONSTANT(T, 113, 487.268001604651932322080970189930074),
266 BOOST_MATH_BIG_CONSTANT(T, 113, 119.953445242335730062471193124820659),
267 };
268 static const T Q_4_8[] = {
269 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
270 BOOST_MATH_BIG_CONSTANT(T, 113, 12.4720855670474488978638945855932398),
271 BOOST_MATH_BIG_CONSTANT(T, 113, 78.6093129753298570701376952709727391),
272 BOOST_MATH_BIG_CONSTANT(T, 113, 307.470246050318322489781182863190127),
273 BOOST_MATH_BIG_CONSTANT(T, 113, 805.140686101151538537565264188630079),
274 BOOST_MATH_BIG_CONSTANT(T, 113, 1439.12019760292146454787601409644413),
275 BOOST_MATH_BIG_CONSTANT(T, 113, 1735.6105285756048831268586001383127),
276 BOOST_MATH_BIG_CONSTANT(T, 113, 1348.32500712856328019355198611280536),
277 BOOST_MATH_BIG_CONSTANT(T, 113, 607.225985860570846699704222144650563),
278 BOOST_MATH_BIG_CONSTANT(T, 113, 119.952317857277045332558673164517227),
279 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000140165918355036060868680809129436084),
280 };
281
282 // Maximum Deviation Found: 2.867e-035
283 // Expected Error Term : 2.866e-035
284 // Maximum Relative Change in Control Points : 2.662e-004
285 static const T P_8_16[] = {
286 BOOST_MATH_BIG_CONSTANT(T, 113, -0.184828315274146610610872315609837439e-19),
287 BOOST_MATH_BIG_CONSTANT(T, 113, 0.500000000000000004122475157735807738),
288 BOOST_MATH_BIG_CONSTANT(T, 113, 3.02533865247313349284875558880415875),
289 BOOST_MATH_BIG_CONSTANT(T, 113, 13.5995927517457371243039532492642734),
290 BOOST_MATH_BIG_CONSTANT(T, 113, 35.3132224283087906757037999452941588),
291 BOOST_MATH_BIG_CONSTANT(T, 113, 67.1639424550714159157603179911505619),
292 BOOST_MATH_BIG_CONSTANT(T, 113, 83.5767733658513967581959839367419891),
293 BOOST_MATH_BIG_CONSTANT(T, 113, 71.073491212235705900866411319363501),
294 BOOST_MATH_BIG_CONSTANT(T, 113, 35.8621515614725564575893663483998663),
295 BOOST_MATH_BIG_CONSTANT(T, 113, 8.72152231639983491987779743154333318),
296 };
297 static const T Q_8_16[] = {
298 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
299 BOOST_MATH_BIG_CONSTANT(T, 113, 5.71734397161293452310624822415866372),
300 BOOST_MATH_BIG_CONSTANT(T, 113, 25.293404179620438179337103263274815),
301 BOOST_MATH_BIG_CONSTANT(T, 113, 62.2619767967468199111077640625328469),
302 BOOST_MATH_BIG_CONSTANT(T, 113, 113.955048909238993473389714972250235),
303 BOOST_MATH_BIG_CONSTANT(T, 113, 130.807138328938966981862203944329408),
304 BOOST_MATH_BIG_CONSTANT(T, 113, 102.423146902337654110717764213057753),
305 BOOST_MATH_BIG_CONSTANT(T, 113, 44.0424772805245202514468199602123565),
306 BOOST_MATH_BIG_CONSTANT(T, 113, 8.89898032477904072082994913461386099),
307 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0296627336872039988632793863671456398),
308 };
309 // Maximum Deviation Found: 1.079e-035
310 // Expected Error Term : -1.079e-035
311 // Maximum Relative Change in Control Points : 7.884e-003
312 static const T P_16_inf[] = {
313 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0),
314 BOOST_MATH_BIG_CONSTANT(T, 113, 0.500000000000000000000000000000087317),
315 BOOST_MATH_BIG_CONSTANT(T, 113, 0.345625669885456215194494735902663968),
316 BOOST_MATH_BIG_CONSTANT(T, 113, 9.62895499360842232127552650044647769),
317 BOOST_MATH_BIG_CONSTANT(T, 113, 3.5936085382439026269301003761320812),
318 BOOST_MATH_BIG_CONSTANT(T, 113, 49.459599118438883265036646019410669),
319 BOOST_MATH_BIG_CONSTANT(T, 113, 7.77519237321893917784735690560496607),
320 BOOST_MATH_BIG_CONSTANT(T, 113, 74.4536074488178075948642351179304121),
321 BOOST_MATH_BIG_CONSTANT(T, 113, 2.75209340397069050436806159297952699),
322 BOOST_MATH_BIG_CONSTANT(T, 113, 23.9292359711471667884504840186561598),
323 };
324 static const T Q_16_inf[] = {
325 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
326 BOOST_MATH_BIG_CONSTANT(T, 113, 0.357918006437579097055656138920742037),
327 BOOST_MATH_BIG_CONSTANT(T, 113, 19.1386039850709849435325005484512944),
328 BOOST_MATH_BIG_CONSTANT(T, 113, 0.874349081464143606016221431763364517),
329 BOOST_MATH_BIG_CONSTANT(T, 113, 98.6516097434855572678195488061432509),
330 BOOST_MATH_BIG_CONSTANT(T, 113, -16.1051972833382893468655223662534306),
331 BOOST_MATH_BIG_CONSTANT(T, 113, 154.316860216253720989145047141653727),
332 BOOST_MATH_BIG_CONSTANT(T, 113, -40.2026880424378986053105969312264534),
333 BOOST_MATH_BIG_CONSTANT(T, 113, 60.1679136674264778074736441126810223),
334 BOOST_MATH_BIG_CONSTANT(T, 113, -13.3414844622256422644504472438320114),
335 BOOST_MATH_BIG_CONSTANT(T, 113, 2.53795636200649908779512969030363442),
336 };
337
338 if(x <= 2)
339 {
340 return (2 + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x);
341 }
342 else if(x <= 4)
343 {
344 return (y_offset_2_4 + boost::math::tools::evaluate_polynomial(P_2_4, x) / tools::evaluate_polynomial(Q_2_4, x)) / (x * x);
345 }
346 else if(x <= 8)
347 {
348 T y = 1 / x;
349 return (1 + tools::evaluate_polynomial(P_4_8, y) / tools::evaluate_polynomial(Q_4_8, y)) / x;
350 }
351 else if(x <= 16)
352 {
353 T y = 1 / x;
354 return (1 + tools::evaluate_polynomial(P_8_16, y) / tools::evaluate_polynomial(Q_8_16, y)) / x;
355 }
356 T y = 1 / x;
357 return (1 + tools::evaluate_polynomial(P_16_inf, y) / tools::evaluate_polynomial(Q_16_inf, y)) / x;
358}
359
360template <class T, class Tag, class Policy>
361T trigamma_imp(T x, const Tag* t, const Policy& pol)
362{
363 //
364 // This handles reflection of negative arguments, and all our
365 // error handling, then forwards to the T-specific approximation.
366 //
367 BOOST_MATH_STD_USING // ADL of std functions.
368
369 T result = 0;
370 //
371 // Check for negative arguments and use reflection:
372 //
373 if(x <= 0)
374 {
375 // Reflect:
376 T z = 1 - x;
377 // Argument reduction for tan:
378 if(floor(x) == x)
379 {
380 return policies::raise_pole_error<T>("boost::math::trigamma<%1%>(%1%)", 0, (1-x), pol);
381 }
382 T s = fabs(x) < fabs(z) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(z, pol);
383 return -trigamma_imp(z, t, pol) + boost::math::pow<2>(constants::pi<T>()) / (s * s);
384 }
385 if(x < 1)
386 {
387 result = 1 / (x * x);
388 x += 1;
389 }
390 return result + trigamma_prec(x, t, pol);
391}
392
393template <class T, class Policy>
394T trigamma_imp(T x, const boost::integral_constant<int, 0>*, const Policy& pol)
395{
396 return polygamma_imp(1, x, pol);
397}
398//
399// Initializer: ensure all our constants are initialized prior to the first call of main:
400//
401template <class T, class Policy>
402struct trigamma_initializer
403{
404 struct init
405 {
406 init()
407 {
408 typedef typename policies::precision<T, Policy>::type precision_type;
409 do_init(boost::integral_constant<bool, precision_type::value && (precision_type::value <= 113)>());
410 }
411 void do_init(const boost::true_type&)
412 {
413 boost::math::trigamma(T(2.5), Policy());
414 }
415 void do_init(const boost::false_type&){}
416 void force_instantiate()const{}
417 };
418 static const init initializer;
419 static void force_instantiate()
420 {
421 initializer.force_instantiate();
422 }
423};
424
425template <class T, class Policy>
426const typename trigamma_initializer<T, Policy>::init trigamma_initializer<T, Policy>::initializer;
427
428} // namespace detail
429
430template <class T, class Policy>
431inline typename tools::promote_args<T>::type
432 trigamma(T x, const Policy&)
433{
434 typedef typename tools::promote_args<T>::type result_type;
435 typedef typename policies::evaluation<result_type, Policy>::type value_type;
436 typedef typename policies::precision<T, Policy>::type precision_type;
437 typedef boost::integral_constant<int,
438 precision_type::value <= 0 ? 0 :
439 precision_type::value <= 53 ? 53 :
440 precision_type::value <= 64 ? 64 :
441 precision_type::value <= 113 ? 113 : 0
442 > tag_type;
443 typedef typename policies::normalise<
444 Policy,
445 policies::promote_float<false>,
446 policies::promote_double<false>,
447 policies::discrete_quantile<>,
448 policies::assert_undefined<> >::type forwarding_policy;
449
450 // Force initialization of constants:
451 detail::trigamma_initializer<value_type, forwarding_policy>::force_instantiate();
452
453 return policies::checked_narrowing_cast<result_type, Policy>(detail::trigamma_imp(
454 static_cast<value_type>(x),
455 static_cast<const tag_type*>(0), forwarding_policy()), "boost::math::trigamma<%1%>(%1%)");
456}
457
458template <class T>
459inline typename tools::promote_args<T>::type
460 trigamma(T x)
461{
462 return trigamma(x, policies::policy<>());
463}
464
465} // namespace math
466} // namespace boost
467#endif
468
469

source code of include/boost/math/special_functions/trigamma.hpp