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author Chris Cannam
date Mon, 07 Sep 2015 11:13:41 +0100
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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 namespace boost{
24 namespace math{
25 namespace detail{
26
27 template<class T, class Policy>
28 T polygamma_imp(const int n, T x, const Policy &pol);
29
30 template <class T, class Policy>
31 T trigamma_prec(T x, const mpl::int_<53>*, const Policy&)
32 {
33 // Max error in interpolated form: 3.736e-017
34 static const T offset = BOOST_MATH_BIG_CONSTANT(T, 53, 2.1093254089355469);
35 static const T P_1_2[] = {
36 BOOST_MATH_BIG_CONSTANT(T, 53, -1.1093280605946045),
37 BOOST_MATH_BIG_CONSTANT(T, 53, -3.8310674472619321),
38 BOOST_MATH_BIG_CONSTANT(T, 53, -3.3703848401898283),
39 BOOST_MATH_BIG_CONSTANT(T, 53, 0.28080574467981213),
40 BOOST_MATH_BIG_CONSTANT(T, 53, 1.6638069578676164),
41 BOOST_MATH_BIG_CONSTANT(T, 53, 0.64468386819102836),
42 };
43 static const T Q_1_2[] = {
44 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
45 BOOST_MATH_BIG_CONSTANT(T, 53, 3.4535389668541151),
46 BOOST_MATH_BIG_CONSTANT(T, 53, 4.5208926987851437),
47 BOOST_MATH_BIG_CONSTANT(T, 53, 2.7012734178351534),
48 BOOST_MATH_BIG_CONSTANT(T, 53, 0.64468798399785611),
49 BOOST_MATH_BIG_CONSTANT(T, 53, -0.20314516859987728e-6),
50 };
51 // Max error in interpolated form: 1.159e-017
52 static const T P_2_4[] = {
53 BOOST_MATH_BIG_CONSTANT(T, 53, -0.13803835004508849e-7),
54 BOOST_MATH_BIG_CONSTANT(T, 53, 0.50000049158540261),
55 BOOST_MATH_BIG_CONSTANT(T, 53, 1.6077979838469348),
56 BOOST_MATH_BIG_CONSTANT(T, 53, 2.5645435828098254),
57 BOOST_MATH_BIG_CONSTANT(T, 53, 2.0534873203680393),
58 BOOST_MATH_BIG_CONSTANT(T, 53, 0.74566981111565923),
59 };
60 static const T Q_2_4[] = {
61 BOOST_MATH_BIG_CONSTANT(T, 53, 1.0),
62 BOOST_MATH_BIG_CONSTANT(T, 53, 2.8822787662376169),
63 BOOST_MATH_BIG_CONSTANT(T, 53, 4.1681660554090917),
64 BOOST_MATH_BIG_CONSTANT(T, 53, 2.7853527819234466),
65 BOOST_MATH_BIG_CONSTANT(T, 53, 0.74967671848044792),
66 BOOST_MATH_BIG_CONSTANT(T, 53, -0.00057069112416246805),
67 };
68 // Maximum Deviation Found: 6.896e-018
69 // Expected Error Term : -6.895e-018
70 // Maximum Relative Change in Control Points : 8.497e-004
71 static const T P_4_inf[] = {
72 0.68947581948701249e-17L,
73 0.49999999999998975L,
74 1.0177274392923795L,
75 2.498208511343429L,
76 2.1921221359427595L,
77 1.5897035272532764L,
78 0.40154388356961734L,
79 };
80 static const T Q_4_inf[] = {
81 1.0L,
82 1.7021215452463932L,
83 4.4290431747556469L,
84 2.9745631894384922L,
85 2.3013614809773616L,
86 0.28360399799075752L,
87 0.022892987908906897L,
88 };
89
90 if(x <= 2)
91 {
92 return (offset + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x);
93 }
94 else if(x <= 4)
95 {
96 T y = 1 / x;
97 return (1 + tools::evaluate_polynomial(P_2_4, y) / tools::evaluate_polynomial(Q_2_4, y)) / x;
98 }
99 T y = 1 / x;
100 return (1 + tools::evaluate_polynomial(P_4_inf, y) / tools::evaluate_polynomial(Q_4_inf, y)) / x;
101 }
102
103 template <class T, class Policy>
104 T trigamma_prec(T x, const mpl::int_<64>*, const Policy&)
105 {
106 // Max error in interpolated form: 1.178e-020
107 static const T offset_1_2 = BOOST_MATH_BIG_CONSTANT(T, 64, 2.109325408935546875);
108 static const T P_1_2[] = {
109 BOOST_MATH_BIG_CONSTANT(T, 64, -1.10932535608960258341),
110 BOOST_MATH_BIG_CONSTANT(T, 64, -4.18793841543017129052),
111 BOOST_MATH_BIG_CONSTANT(T, 64, -4.63865531898487734531),
112 BOOST_MATH_BIG_CONSTANT(T, 64, -0.919832884430500908047),
113 BOOST_MATH_BIG_CONSTANT(T, 64, 1.68074038333180423012),
114 BOOST_MATH_BIG_CONSTANT(T, 64, 1.21172611429185622377),
115 BOOST_MATH_BIG_CONSTANT(T, 64, 0.259635673503366427284),
116 };
117 static const T Q_1_2[] = {
118 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
119 BOOST_MATH_BIG_CONSTANT(T, 64, 3.77521119359546982995),
120 BOOST_MATH_BIG_CONSTANT(T, 64, 5.664338024578956321),
121 BOOST_MATH_BIG_CONSTANT(T, 64, 4.25995134879278028361),
122 BOOST_MATH_BIG_CONSTANT(T, 64, 1.62956638448940402182),
123 BOOST_MATH_BIG_CONSTANT(T, 64, 0.259635512844691089868),
124 BOOST_MATH_BIG_CONSTANT(T, 64, 0.629642219810618032207e-8),
125 };
126 // Max error in interpolated form: 3.912e-020
127 static const T P_2_8[] = {
128 BOOST_MATH_BIG_CONSTANT(T, 64, -0.387540035162952880976e-11),
129 BOOST_MATH_BIG_CONSTANT(T, 64, 0.500000000276430504),
130 BOOST_MATH_BIG_CONSTANT(T, 64, 3.21926880986360957306),
131 BOOST_MATH_BIG_CONSTANT(T, 64, 10.2550347708483445775),
132 BOOST_MATH_BIG_CONSTANT(T, 64, 18.9002075150709144043),
133 BOOST_MATH_BIG_CONSTANT(T, 64, 21.0357215832399705625),
134 BOOST_MATH_BIG_CONSTANT(T, 64, 13.4346512182925923978),
135 BOOST_MATH_BIG_CONSTANT(T, 64, 3.98656291026448279118),
136 };
137 static const T Q_2_8[] = {
138 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
139 BOOST_MATH_BIG_CONSTANT(T, 64, 6.10520430478613667724),
140 BOOST_MATH_BIG_CONSTANT(T, 64, 18.475001060603645512),
141 BOOST_MATH_BIG_CONSTANT(T, 64, 31.7087534567758405638),
142 BOOST_MATH_BIG_CONSTANT(T, 64, 31.908814523890465398),
143 BOOST_MATH_BIG_CONSTANT(T, 64, 17.4175479039227084798),
144 BOOST_MATH_BIG_CONSTANT(T, 64, 3.98749106958394941276),
145 BOOST_MATH_BIG_CONSTANT(T, 64, -0.000115917322224411128566),
146 };
147 // Maximum Deviation Found: 2.635e-020
148 // Expected Error Term : 2.635e-020
149 // Maximum Relative Change in Control Points : 1.791e-003
150 static const T P_8_inf[] = {
151 BOOST_MATH_BIG_CONSTANT(T, 64, -0.263527875092466899848e-19),
152 BOOST_MATH_BIG_CONSTANT(T, 64, 0.500000000000000058145),
153 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0730121433777364138677),
154 BOOST_MATH_BIG_CONSTANT(T, 64, 1.94505878379957149534),
155 BOOST_MATH_BIG_CONSTANT(T, 64, 0.0517092358874932620529),
156 BOOST_MATH_BIG_CONSTANT(T, 64, 1.07995383547483921121),
157 };
158 static const T Q_8_inf[] = {
159 BOOST_MATH_BIG_CONSTANT(T, 64, 1.0),
160 BOOST_MATH_BIG_CONSTANT(T, 64, -0.187309046577818095504),
161 BOOST_MATH_BIG_CONSTANT(T, 64, 3.95255391645238842975),
162 BOOST_MATH_BIG_CONSTANT(T, 64, -1.14743283327078949087),
163 BOOST_MATH_BIG_CONSTANT(T, 64, 2.52989799376344914499),
164 BOOST_MATH_BIG_CONSTANT(T, 64, -0.627414303172402506396),
165 BOOST_MATH_BIG_CONSTANT(T, 64, 0.141554248216425512536),
166 };
167
168 if(x <= 2)
169 {
170 return (offset_1_2 + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x);
171 }
172 else if(x <= 8)
173 {
174 T y = 1 / x;
175 return (1 + tools::evaluate_polynomial(P_2_8, y) / tools::evaluate_polynomial(Q_2_8, y)) / x;
176 }
177 T y = 1 / x;
178 return (1 + tools::evaluate_polynomial(P_8_inf, y) / tools::evaluate_polynomial(Q_8_inf, y)) / x;
179 }
180
181 template <class T, class Policy>
182 T trigamma_prec(T x, const mpl::int_<113>*, const Policy&)
183 {
184 // Max error in interpolated form: 1.916e-035
185
186 static const T P_1_2[] = {
187 BOOST_MATH_BIG_CONSTANT(T, 113, -0.999999999999999082554457936871832533),
188 BOOST_MATH_BIG_CONSTANT(T, 113, -4.71237311120865266379041700054847734),
189 BOOST_MATH_BIG_CONSTANT(T, 113, -7.94125711970499027763789342500817316),
190 BOOST_MATH_BIG_CONSTANT(T, 113, -5.74657746697664735258222071695644535),
191 BOOST_MATH_BIG_CONSTANT(T, 113, -0.404213349456398905981223965160595687),
192 BOOST_MATH_BIG_CONSTANT(T, 113, 2.47877781178642876561595890095758896),
193 BOOST_MATH_BIG_CONSTANT(T, 113, 2.07714151702455125992166949812126433),
194 BOOST_MATH_BIG_CONSTANT(T, 113, 0.858877899162360138844032265418028567),
195 BOOST_MATH_BIG_CONSTANT(T, 113, 0.20499222604410032375789018837922397),
196 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0272103140348194747360175268778415049),
197 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0015764849020876949848954081173520686),
198 };
199 static const T Q_1_2[] = {
200 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
201 BOOST_MATH_BIG_CONSTANT(T, 113, 4.71237311120863419878375031457715223),
202 BOOST_MATH_BIG_CONSTANT(T, 113, 9.58619118655339853449127952145877467),
203 BOOST_MATH_BIG_CONSTANT(T, 113, 11.0940067269829372437561421279054968),
204 BOOST_MATH_BIG_CONSTANT(T, 113, 8.09075424749327792073276309969037885),
205 BOOST_MATH_BIG_CONSTANT(T, 113, 3.87705890159891405185343806884451286),
206 BOOST_MATH_BIG_CONSTANT(T, 113, 1.22758678701914477836330837816976782),
207 BOOST_MATH_BIG_CONSTANT(T, 113, 0.249092040606385004109672077814668716),
208 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0295750413900655597027079600025569048),
209 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00157648490200498142247694709728858139),
210 BOOST_MATH_BIG_CONSTANT(T, 113, 0.161264050344059471721062360645432809e-14),
211 };
212
213 // Max error in interpolated form: 8.958e-035
214 static const T P_2_4[] = {
215 BOOST_MATH_BIG_CONSTANT(T, 113, -2.55843734739907925764326773972215085),
216 BOOST_MATH_BIG_CONSTANT(T, 113, -12.2830208240542011967952466273455887),
217 BOOST_MATH_BIG_CONSTANT(T, 113, -23.9195022162767993526575786066414403),
218 BOOST_MATH_BIG_CONSTANT(T, 113, -24.9256431504823483094158828285470862),
219 BOOST_MATH_BIG_CONSTANT(T, 113, -14.7979122765478779075108064826412285),
220 BOOST_MATH_BIG_CONSTANT(T, 113, -4.46654453928610666393276765059122272),
221 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0191439033405649675717082465687845002),
222 BOOST_MATH_BIG_CONSTANT(T, 113, 0.515412052554351265708917209749037352),
223 BOOST_MATH_BIG_CONSTANT(T, 113, 0.195378348786064304378247325360320038),
224 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0334761282624174313035014426794245393),
225 BOOST_MATH_BIG_CONSTANT(T, 113, 0.002373665205942206348500250056602687),
226 };
227 static const T Q_2_4[] = {
228 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
229 BOOST_MATH_BIG_CONSTANT(T, 113, 4.80098558454419907830670928248659245),
230 BOOST_MATH_BIG_CONSTANT(T, 113, 9.99220727843170133895059300223445265),
231 BOOST_MATH_BIG_CONSTANT(T, 113, 11.8896146167631330735386697123464976),
232 BOOST_MATH_BIG_CONSTANT(T, 113, 8.96613256683809091593793565879092581),
233 BOOST_MATH_BIG_CONSTANT(T, 113, 4.47254136149624110878909334574485751),
234 BOOST_MATH_BIG_CONSTANT(T, 113, 1.48600982028196527372434773913633152),
235 BOOST_MATH_BIG_CONSTANT(T, 113, 0.319570735766764237068541501137990078),
236 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0407358345787680953107374215319322066),
237 BOOST_MATH_BIG_CONSTANT(T, 113, 0.00237366520593271641375755486420859837),
238 BOOST_MATH_BIG_CONSTANT(T, 113, 0.239554887903526152679337256236302116e-15),
239 BOOST_MATH_BIG_CONSTANT(T, 113, -0.294749244740618656265237072002026314e-17),
240 };
241
242 static const T y_offset_2_4 = BOOST_MATH_BIG_CONSTANT(T, 113, 3.558437347412109375);
243
244 // Max error in interpolated form: 4.319e-035
245 static const T P_4_8[] = {
246 BOOST_MATH_BIG_CONSTANT(T, 113, 0.166626112697021464248967707021688845e-16),
247 BOOST_MATH_BIG_CONSTANT(T, 113, 0.499999999999997739552090249208808197),
248 BOOST_MATH_BIG_CONSTANT(T, 113, 6.40270945019053817915772473771553187),
249 BOOST_MATH_BIG_CONSTANT(T, 113, 41.3833374155000608013677627389343329),
250 BOOST_MATH_BIG_CONSTANT(T, 113, 166.803341854562809335667241074035245),
251 BOOST_MATH_BIG_CONSTANT(T, 113, 453.39964786925369319960722793414521),
252 BOOST_MATH_BIG_CONSTANT(T, 113, 851.153712317697055375935433362983944),
253 BOOST_MATH_BIG_CONSTANT(T, 113, 1097.70657567285059133109286478004458),
254 BOOST_MATH_BIG_CONSTANT(T, 113, 938.431232478455316020076349367632922),
255 BOOST_MATH_BIG_CONSTANT(T, 113, 487.268001604651932322080970189930074),
256 BOOST_MATH_BIG_CONSTANT(T, 113, 119.953445242335730062471193124820659),
257 };
258 static const T Q_4_8[] = {
259 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
260 BOOST_MATH_BIG_CONSTANT(T, 113, 12.4720855670474488978638945855932398),
261 BOOST_MATH_BIG_CONSTANT(T, 113, 78.6093129753298570701376952709727391),
262 BOOST_MATH_BIG_CONSTANT(T, 113, 307.470246050318322489781182863190127),
263 BOOST_MATH_BIG_CONSTANT(T, 113, 805.140686101151538537565264188630079),
264 BOOST_MATH_BIG_CONSTANT(T, 113, 1439.12019760292146454787601409644413),
265 BOOST_MATH_BIG_CONSTANT(T, 113, 1735.6105285756048831268586001383127),
266 BOOST_MATH_BIG_CONSTANT(T, 113, 1348.32500712856328019355198611280536),
267 BOOST_MATH_BIG_CONSTANT(T, 113, 607.225985860570846699704222144650563),
268 BOOST_MATH_BIG_CONSTANT(T, 113, 119.952317857277045332558673164517227),
269 BOOST_MATH_BIG_CONSTANT(T, 113, 0.000140165918355036060868680809129436084),
270 };
271
272 // Maximum Deviation Found: 2.867e-035
273 // Expected Error Term : 2.866e-035
274 // Maximum Relative Change in Control Points : 2.662e-004
275 static const T P_8_16[] = {
276 BOOST_MATH_BIG_CONSTANT(T, 113, -0.184828315274146610610872315609837439e-19),
277 BOOST_MATH_BIG_CONSTANT(T, 113, 0.500000000000000004122475157735807738),
278 BOOST_MATH_BIG_CONSTANT(T, 113, 3.02533865247313349284875558880415875),
279 BOOST_MATH_BIG_CONSTANT(T, 113, 13.5995927517457371243039532492642734),
280 BOOST_MATH_BIG_CONSTANT(T, 113, 35.3132224283087906757037999452941588),
281 BOOST_MATH_BIG_CONSTANT(T, 113, 67.1639424550714159157603179911505619),
282 BOOST_MATH_BIG_CONSTANT(T, 113, 83.5767733658513967581959839367419891),
283 BOOST_MATH_BIG_CONSTANT(T, 113, 71.073491212235705900866411319363501),
284 BOOST_MATH_BIG_CONSTANT(T, 113, 35.8621515614725564575893663483998663),
285 BOOST_MATH_BIG_CONSTANT(T, 113, 8.72152231639983491987779743154333318),
286 };
287 static const T Q_8_16[] = {
288 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
289 BOOST_MATH_BIG_CONSTANT(T, 113, 5.71734397161293452310624822415866372),
290 BOOST_MATH_BIG_CONSTANT(T, 113, 25.293404179620438179337103263274815),
291 BOOST_MATH_BIG_CONSTANT(T, 113, 62.2619767967468199111077640625328469),
292 BOOST_MATH_BIG_CONSTANT(T, 113, 113.955048909238993473389714972250235),
293 BOOST_MATH_BIG_CONSTANT(T, 113, 130.807138328938966981862203944329408),
294 BOOST_MATH_BIG_CONSTANT(T, 113, 102.423146902337654110717764213057753),
295 BOOST_MATH_BIG_CONSTANT(T, 113, 44.0424772805245202514468199602123565),
296 BOOST_MATH_BIG_CONSTANT(T, 113, 8.89898032477904072082994913461386099),
297 BOOST_MATH_BIG_CONSTANT(T, 113, -0.0296627336872039988632793863671456398),
298 };
299 // Maximum Deviation Found: 1.079e-035
300 // Expected Error Term : -1.079e-035
301 // Maximum Relative Change in Control Points : 7.884e-003
302 static const T P_16_inf[] = {
303 BOOST_MATH_BIG_CONSTANT(T, 113, 0.0),
304 BOOST_MATH_BIG_CONSTANT(T, 113, 0.500000000000000000000000000000087317),
305 BOOST_MATH_BIG_CONSTANT(T, 113, 0.345625669885456215194494735902663968),
306 BOOST_MATH_BIG_CONSTANT(T, 113, 9.62895499360842232127552650044647769),
307 BOOST_MATH_BIG_CONSTANT(T, 113, 3.5936085382439026269301003761320812),
308 BOOST_MATH_BIG_CONSTANT(T, 113, 49.459599118438883265036646019410669),
309 BOOST_MATH_BIG_CONSTANT(T, 113, 7.77519237321893917784735690560496607),
310 BOOST_MATH_BIG_CONSTANT(T, 113, 74.4536074488178075948642351179304121),
311 BOOST_MATH_BIG_CONSTANT(T, 113, 2.75209340397069050436806159297952699),
312 BOOST_MATH_BIG_CONSTANT(T, 113, 23.9292359711471667884504840186561598),
313 };
314 static const T Q_16_inf[] = {
315 BOOST_MATH_BIG_CONSTANT(T, 113, 1.0),
316 BOOST_MATH_BIG_CONSTANT(T, 113, 0.357918006437579097055656138920742037),
317 BOOST_MATH_BIG_CONSTANT(T, 113, 19.1386039850709849435325005484512944),
318 BOOST_MATH_BIG_CONSTANT(T, 113, 0.874349081464143606016221431763364517),
319 BOOST_MATH_BIG_CONSTANT(T, 113, 98.6516097434855572678195488061432509),
320 BOOST_MATH_BIG_CONSTANT(T, 113, -16.1051972833382893468655223662534306),
321 BOOST_MATH_BIG_CONSTANT(T, 113, 154.316860216253720989145047141653727),
322 BOOST_MATH_BIG_CONSTANT(T, 113, -40.2026880424378986053105969312264534),
323 BOOST_MATH_BIG_CONSTANT(T, 113, 60.1679136674264778074736441126810223),
324 BOOST_MATH_BIG_CONSTANT(T, 113, -13.3414844622256422644504472438320114),
325 BOOST_MATH_BIG_CONSTANT(T, 113, 2.53795636200649908779512969030363442),
326 };
327
328 if(x <= 2)
329 {
330 return (2 + boost::math::tools::evaluate_polynomial(P_1_2, x) / tools::evaluate_polynomial(Q_1_2, x)) / (x * x);
331 }
332 else if(x <= 4)
333 {
334 return (y_offset_2_4 + boost::math::tools::evaluate_polynomial(P_2_4, x) / tools::evaluate_polynomial(Q_2_4, x)) / (x * x);
335 }
336 else if(x <= 8)
337 {
338 T y = 1 / x;
339 return (1 + tools::evaluate_polynomial(P_4_8, y) / tools::evaluate_polynomial(Q_4_8, y)) / x;
340 }
341 else if(x <= 16)
342 {
343 T y = 1 / x;
344 return (1 + tools::evaluate_polynomial(P_8_16, y) / tools::evaluate_polynomial(Q_8_16, y)) / x;
345 }
346 T y = 1 / x;
347 return (1 + tools::evaluate_polynomial(P_16_inf, y) / tools::evaluate_polynomial(Q_16_inf, y)) / x;
348 }
349
350 template <class T, class Tag, class Policy>
351 T trigamma_imp(T x, const Tag* t, const Policy& pol)
352 {
353 //
354 // This handles reflection of negative arguments, and all our
355 // error handling, then forwards to the T-specific approximation.
356 //
357 BOOST_MATH_STD_USING // ADL of std functions.
358
359 T result = 0;
360 //
361 // Check for negative arguments and use reflection:
362 //
363 if(x <= 0)
364 {
365 // Reflect:
366 T z = 1 - x;
367 // Argument reduction for tan:
368 if(floor(x) == x)
369 {
370 return policies::raise_pole_error<T>("boost::math::trigamma<%1%>(%1%)", 0, (1-x), pol);
371 }
372 T s = fabs(x) < fabs(z) ? boost::math::sin_pi(x, pol) : boost::math::sin_pi(z, pol);
373 return -trigamma_imp(z, t, pol) + boost::math::pow<2>(constants::pi<T>()) / (s * s);
374 }
375 if(x < 1)
376 {
377 result = 1 / (x * x);
378 x += 1;
379 }
380 return result + trigamma_prec(x, t, pol);
381 }
382
383 template <class T, class Policy>
384 T trigamma_imp(T x, const mpl::int_<0>*, const Policy& pol)
385 {
386 return polygamma_imp(1, x, pol);
387 }
388 //
389 // Initializer: ensure all our constants are initialized prior to the first call of main:
390 //
391 template <class T, class Policy>
392 struct trigamma_initializer
393 {
394 struct init
395 {
396 init()
397 {
398 typedef typename policies::precision<T, Policy>::type precision_type;
399 do_init(mpl::bool_<precision_type::value && (precision_type::value <= 113)>());
400 }
401 void do_init(const mpl::true_&)
402 {
403 boost::math::trigamma(T(2.5), Policy());
404 }
405 void do_init(const mpl::false_&){}
406 void force_instantiate()const{}
407 };
408 static const init initializer;
409 static void force_instantiate()
410 {
411 initializer.force_instantiate();
412 }
413 };
414
415 template <class T, class Policy>
416 const typename trigamma_initializer<T, Policy>::init trigamma_initializer<T, Policy>::initializer;
417
418 } // namespace detail
419
420 template <class T, class Policy>
421 inline typename tools::promote_args<T>::type
422 trigamma(T x, const Policy&)
423 {
424 typedef typename tools::promote_args<T>::type result_type;
425 typedef typename policies::evaluation<result_type, Policy>::type value_type;
426 typedef typename policies::precision<T, Policy>::type precision_type;
427 typedef typename mpl::if_<
428 mpl::or_<
429 mpl::less_equal<precision_type, mpl::int_<0> >,
430 mpl::greater<precision_type, mpl::int_<114> >
431 >,
432 mpl::int_<0>,
433 typename mpl::if_<
434 mpl::less<precision_type, mpl::int_<54> >,
435 mpl::int_<53>,
436 typename mpl::if_<
437 mpl::less<precision_type, mpl::int_<65> >,
438 mpl::int_<64>,
439 mpl::int_<113>
440 >::type
441 >::type
442 >::type tag_type;
443
444 typedef typename policies::normalise<
445 Policy,
446 policies::promote_float<false>,
447 policies::promote_double<false>,
448 policies::discrete_quantile<>,
449 policies::assert_undefined<> >::type forwarding_policy;
450
451 // Force initialization of constants:
452 detail::trigamma_initializer<value_type, forwarding_policy>::force_instantiate();
453
454 return policies::checked_narrowing_cast<result_type, Policy>(detail::trigamma_imp(
455 static_cast<value_type>(x),
456 static_cast<const tag_type*>(0), forwarding_policy()), "boost::math::trigamma<%1%>(%1%)");
457 }
458
459 template <class T>
460 inline typename tools::promote_args<T>::type
461 trigamma(T x)
462 {
463 return trigamma(x, policies::policy<>());
464 }
465
466 } // namespace math
467 } // namespace boost
468 #endif
469