Mercurial > hg > vamp-build-and-test

comparison DEPENDENCIES/generic/include/boost/math/special_functions/lanczos.hpp @ 16:2665513ce2d3

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Add boost headers
author Chris Cannam
date Tue, 05 Aug 2014 11:11:38 +0100
parents
children c530137014c0
comparison
equal deleted inserted replaced
15:663ca0da4350 16:2665513ce2d3
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_SPECIAL_FUNCTIONS_LANCZOS
7 #define BOOST_MATH_SPECIAL_FUNCTIONS_LANCZOS
8
9 #ifdef _MSC_VER
10 #pragma once
11 #endif
12
13 #include <boost/config.hpp>
14 #include <boost/math/tools/big_constant.hpp>
15 #include <boost/mpl/if.hpp>
16 #include <boost/limits.hpp>
17 #include <boost/cstdint.hpp>
18 #include <boost/math/tools/rational.hpp>
19 #include <boost/math/policies/policy.hpp>
20 #include <boost/mpl/less_equal.hpp>
21
22 #include <limits.h>
23
24 namespace boost{ namespace math{ namespace lanczos{
25
26 //
27 // Individual lanczos approximations start here.
28 //
29 // Optimal values for G for each N are taken from
30 // http://web.mala.bc.ca/pughg/phdThesis/phdThesis.pdf,
31 // as are the theoretical error bounds.
32 //
33 // Constants calculated using the method described by Godfrey
34 // http://my.fit.edu/~gabdo/gamma.txt and elaborated by Toth at
35 // http://www.rskey.org/gamma.htm using NTL::RR at 1000 bit precision.
36 //
37 // Begin with a small helper to force initialization of constants prior
38 // to main. This makes the constant initialization thread safe, even
39 // when called with a user-defined number type.
40 //
41 template <class Lanczos, class T>
42 struct lanczos_initializer
43 {
44 struct init
45 {
46 init()
47 {
48 T t(1);
49 Lanczos::lanczos_sum(t);
50 Lanczos::lanczos_sum_expG_scaled(t);
51 Lanczos::lanczos_sum_near_1(t);
52 Lanczos::lanczos_sum_near_2(t);
53 Lanczos::g();
54 }
55 void force_instantiate()const{}
56 };
57 static const init initializer;
58 static void force_instantiate()
59 {
60 initializer.force_instantiate();
61 }
62 };
63 template <class Lanczos, class T>
64 typename lanczos_initializer<Lanczos, T>::init const lanczos_initializer<Lanczos, T>::initializer;
65 //
66 // Lanczos Coefficients for N=6 G=5.581
67 // Max experimental error (with arbitary precision arithmetic) 9.516e-12
68 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
69 //
70 struct lanczos6 : public mpl::int_<35>
71 {
72 //
73 // Produces slightly better than float precision when evaluated at
74 // double precision:
75 //
76 template <class T>
77 static T lanczos_sum(const T& z)
78 {
79 lanczos_initializer<lanczos6, T>::force_instantiate(); // Ensure our constants get initialized before main()
80 static const T num[6] = {
81 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 8706.349592549009182288174442774377925882)),
82 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 8523.650341121874633477483696775067709735)),
83 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 3338.029219476423550899999750161289306564)),
84 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 653.6424994294008795995653541449610986791)),
85 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 63.99951844938187085666201263218840287667)),
86 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 2.506628274631006311133031631822390264407))
87 };
88 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint16_t) denom[6] = {
89 static_cast<boost::uint16_t>(0u),
90 static_cast<boost::uint16_t>(24u),
91 static_cast<boost::uint16_t>(50u),
92 static_cast<boost::uint16_t>(35u),
93 static_cast<boost::uint16_t>(10u),
94 static_cast<boost::uint16_t>(1u)
95 };
96 return boost::math::tools::evaluate_rational(num, denom, z);
97 }
98
99 template <class T>
100 static T lanczos_sum_expG_scaled(const T& z)
101 {
102 lanczos_initializer<lanczos6, T>::force_instantiate(); // Ensure our constants get initialized before main()
103 static const T num[6] = {
104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 32.81244541029783471623665933780748627823)),
105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 32.12388941444332003446077108933558534361)),
106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 12.58034729455216106950851080138931470954)),
107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 2.463444478353241423633780693218408889251)),
108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 0.2412010548258800231126240760264822486599)),
109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 0.009446967704539249494420221613134244048319))
110 };
111 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint16_t) denom[6] = {
112 static_cast<boost::uint16_t>(0u),
113 static_cast<boost::uint16_t>(24u),
114 static_cast<boost::uint16_t>(50u),
115 static_cast<boost::uint16_t>(35u),
116 static_cast<boost::uint16_t>(10u),
117 static_cast<boost::uint16_t>(1u)
118 };
119 return boost::math::tools::evaluate_rational(num, denom, z);
120 }
121
122
123 template<class T>
124 static T lanczos_sum_near_1(const T& dz)
125 {
126 lanczos_initializer<lanczos6, T>::force_instantiate(); // Ensure our constants get initialized before main()
127 static const T d[5] = {
128 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 2.044879010930422922760429926121241330235)),
129 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, -2.751366405578505366591317846728753993668)),
130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 1.02282965224225004296750609604264824677)),
131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, -0.09786124911582813985028889636665335893627)),
132 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 0.0009829742267506615183144364420540766510112)),
133 };
134 T result = 0;
135 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
136 {
137 result += (-d[k-1]*dz)/(k*dz + k*k);
138 }
139 return result;
140 }
141
142 template<class T>
143 static T lanczos_sum_near_2(const T& dz)
144 {
145 lanczos_initializer<lanczos6, T>::force_instantiate(); // Ensure our constants get initialized before main()
146 static const T d[5] = {
147 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 5.748142489536043490764289256167080091892)),
148 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, -7.734074268282457156081021756682138251825)),
149 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 2.875167944990511006997713242805893543947)),
150 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, -0.2750873773533504542306766137703788781776)),
151 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 35, 0.002763134585812698552178368447708846850353)),
152 };
153 T result = 0;
154 T z = dz + 2;
155 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
156 {
157 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
158 }
159 return result;
160 }
161
162 static double g(){ return 5.581000000000000405009359383257105946541; }
163 };
164
165 //
166 // Lanczos Coefficients for N=11 G=10.900511
167 // Max experimental error (with arbitary precision arithmetic) 2.16676e-19
168 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
169 //
170 struct lanczos11 : public mpl::int_<60>
171 {
172 //
173 // Produces slightly better than double precision when evaluated at
174 // extended-double precision:
175 //
176 template <class T>
177 static T lanczos_sum(const T& z)
178 {
179 lanczos_initializer<lanczos11, T>::force_instantiate(); // Ensure our constants get initialized before main()
180 static const T num[11] = {
181 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 38474670393.31776828316099004518914832218)),
182 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 36857665043.51950660081971227404959150474)),
183 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 15889202453.72942008945006665994637853242)),
184 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 4059208354.298834770194507810788393801607)),
185 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 680547661.1834733286087695557084801366446)),
186 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 78239755.00312005289816041245285376206263)),
187 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 6246580.776401795264013335510453568106366)),
188 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 341986.3488721347032223777872763188768288)),
189 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 12287.19451182455120096222044424100527629)),
190 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 261.6140441641668190791708576058805625502)),
191 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 2.506628274631000502415573855452633787834))
192 };
193 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint32_t) denom[11] = {
194 static_cast<boost::uint32_t>(0u),
195 static_cast<boost::uint32_t>(362880u),
196 static_cast<boost::uint32_t>(1026576u),
197 static_cast<boost::uint32_t>(1172700u),
198 static_cast<boost::uint32_t>(723680u),
199 static_cast<boost::uint32_t>(269325u),
200 static_cast<boost::uint32_t>(63273u),
201 static_cast<boost::uint32_t>(9450u),
202 static_cast<boost::uint32_t>(870u),
203 static_cast<boost::uint32_t>(45u),
204 static_cast<boost::uint32_t>(1u)
205 };
206 return boost::math::tools::evaluate_rational(num, denom, z);
207 }
208
209 template <class T>
210 static T lanczos_sum_expG_scaled(const T& z)
211 {
212 lanczos_initializer<lanczos11, T>::force_instantiate(); // Ensure our constants get initialized before main()
213 static const T num[11] = {
214 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 709811.662581657956893540610814842699825)),
215 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 679979.847415722640161734319823103390728)),
216 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 293136.785721159725251629480984140341656)),
217 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 74887.5403291467179935942448101441897121)),
218 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 12555.29058241386295096255111537516768137)),
219 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 1443.42992444170669746078056942194198252)),
220 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 115.2419459613734722083208906727972935065)),
221 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 6.30923920573262762719523981992008976989)),
222 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.2266840463022436475495508977579735223818)),
223 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.004826466289237661857584712046231435101741)),
224 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.4624429436045378766270459638520555557321e-4))
225 };
226 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint32_t) denom[11] = {
227 static_cast<boost::uint32_t>(0u),
228 static_cast<boost::uint32_t>(362880u),
229 static_cast<boost::uint32_t>(1026576u),
230 static_cast<boost::uint32_t>(1172700u),
231 static_cast<boost::uint32_t>(723680u),
232 static_cast<boost::uint32_t>(269325u),
233 static_cast<boost::uint32_t>(63273u),
234 static_cast<boost::uint32_t>(9450u),
235 static_cast<boost::uint32_t>(870u),
236 static_cast<boost::uint32_t>(45u),
237 static_cast<boost::uint32_t>(1u)
238 };
239 return boost::math::tools::evaluate_rational(num, denom, z);
240 }
241
242
243 template<class T>
244 static T lanczos_sum_near_1(const T& dz)
245 {
246 lanczos_initializer<lanczos11, T>::force_instantiate(); // Ensure our constants get initialized before main()
247 static const T d[10] = {
248 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 4.005853070677940377969080796551266387954)),
249 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -13.17044315127646469834125159673527183164)),
250 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 17.19146865350790353683895137079288129318)),
251 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -11.36446409067666626185701599196274701126)),
252 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 4.024801119349323770107694133829772634737)),
253 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -0.7445703262078094128346501724255463005006)),
254 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.06513861351917497265045550019547857713172)),
255 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -0.00217899958561830354633560009312512312758)),
256 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.17655204574495137651670832229571934738e-4)),
257 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -0.1036282091079938047775645941885460820853e-7)),
258 };
259 T result = 0;
260 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
261 {
262 result += (-d[k-1]*dz)/(k*dz + k*k);
263 }
264 return result;
265 }
266
267 template<class T>
268 static T lanczos_sum_near_2(const T& dz)
269 {
270 lanczos_initializer<lanczos11, T>::force_instantiate(); // Ensure our constants get initialized before main()
271 static const T d[10] = {
272 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 19.05889633808148715159575716844556056056)),
273 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -62.66183664701721716960978577959655644762)),
274 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 81.7929198065004751699057192860287512027)),
275 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -54.06941772964234828416072865069196553015)),
276 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 19.14904664790693019642068229478769661515)),
277 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -3.542488556926667589704590409095331790317)),
278 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.3099140334815639910894627700232804503017)),
279 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -0.01036716187296241640634252431913030440825)),
280 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, 0.8399926504443119927673843789048514017761e-4)),
281 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 60, -0.493038376656195010308610694048822561263e-7)),
282 };
283 T result = 0;
284 T z = dz + 2;
285 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
286 {
287 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
288 }
289 return result;
290 }
291
292 static double g(){ return 10.90051099999999983936049829935654997826; }
293 };
294
295 //
296 // Lanczos Coefficients for N=13 G=13.144565
297 // Max experimental error (with arbitary precision arithmetic) 9.2213e-23
298 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
299 //
300 struct lanczos13 : public mpl::int_<72>
301 {
302 //
303 // Produces slightly better than extended-double precision when evaluated at
304 // higher precision:
305 //
306 template <class T>
307 static T lanczos_sum(const T& z)
308 {
309 lanczos_initializer<lanczos13, T>::force_instantiate(); // Ensure our constants get initialized before main()
310 static const T num[13] = {
311 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 44012138428004.60895436261759919070125699)),
312 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 41590453358593.20051581730723108131357995)),
313 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 18013842787117.99677796276038389462742949)),
314 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 4728736263475.388896889723995205703970787)),
315 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 837910083628.4046470415724300225777912264)),
316 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 105583707273.4299344907359855510105321192)),
317 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 9701363618.494999493386608345339104922694)),
318 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 654914397.5482052641016767125048538245644)),
319 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 32238322.94213356530668889463945849409184)),
320 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 1128514.219497091438040721811544858643121)),
321 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 26665.79378459858944762533958798805525125)),
322 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 381.8801248632926870394389468349331394196)),
323 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 2.506628274631000502415763426076722427007))
324 };
325 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint32_t) denom[13] = {
326 static_cast<boost::uint32_t>(0u),
327 static_cast<boost::uint32_t>(39916800u),
328 static_cast<boost::uint32_t>(120543840u),
329 static_cast<boost::uint32_t>(150917976u),
330 static_cast<boost::uint32_t>(105258076u),
331 static_cast<boost::uint32_t>(45995730u),
332 static_cast<boost::uint32_t>(13339535u),
333 static_cast<boost::uint32_t>(2637558u),
334 static_cast<boost::uint32_t>(357423u),
335 static_cast<boost::uint32_t>(32670u),
336 static_cast<boost::uint32_t>(1925u),
337 static_cast<boost::uint32_t>(66u),
338 static_cast<boost::uint32_t>(1u)
339 };
340 return boost::math::tools::evaluate_rational(num, denom, z);
341 }
342
343 template <class T>
344 static T lanczos_sum_expG_scaled(const T& z)
345 {
346 lanczos_initializer<lanczos13, T>::force_instantiate(); // Ensure our constants get initialized before main()
347 static const T num[13] = {
348 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 86091529.53418537217994842267760536134841)),
349 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 81354505.17858011242874285785316135398567)),
350 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 35236626.38815461910817650960734605416521)),
351 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 9249814.988024471294683815872977672237195)),
352 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 1639024.216687146960253839656643518985826)),
353 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 206530.8157641225032631778026076868855623)),
354 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 18976.70193530288915698282139308582105936)),
355 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 1281.068909912559479885759622791374106059)),
356 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 63.06093343420234536146194868906771599354)),
357 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 2.207470909792527638222674678171050209691)),
358 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.05216058694613505427476207805814960742102)),
359 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.0007469903808915448316510079585999893674101)),
360 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.4903180573459871862552197089738373164184e-5))
361 };
362 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint32_t) denom[13] = {
363 static_cast<boost::uint32_t>(0u),
364 static_cast<boost::uint32_t>(39916800u),
365 static_cast<boost::uint32_t>(120543840u),
366 static_cast<boost::uint32_t>(150917976u),
367 static_cast<boost::uint32_t>(105258076u),
368 static_cast<boost::uint32_t>(45995730u),
369 static_cast<boost::uint32_t>(13339535u),
370 static_cast<boost::uint32_t>(2637558u),
371 static_cast<boost::uint32_t>(357423u),
372 static_cast<boost::uint32_t>(32670u),
373 static_cast<boost::uint32_t>(1925u),
374 static_cast<boost::uint32_t>(66u),
375 static_cast<boost::uint32_t>(1u)
376 };
377 return boost::math::tools::evaluate_rational(num, denom, z);
378 }
379
380
381 template<class T>
382 static T lanczos_sum_near_1(const T& dz)
383 {
384 lanczos_initializer<lanczos13, T>::force_instantiate(); // Ensure our constants get initialized before main()
385 static const T d[12] = {
386 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 4.832115561461656947793029596285626840312)),
387 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -19.86441536140337740383120735104359034688)),
388 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 33.9927422807443239927197864963170585331)),
389 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -31.41520692249765980987427413991250886138)),
390 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 17.0270866009599345679868972409543597821)),
391 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -5.5077216950865501362506920516723682167)),
392 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 1.037811741948214855286817963800439373362)),
393 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -0.106640468537356182313660880481398642811)),
394 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.005276450526660653288757565778182586742831)),
395 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -0.0001000935625597121545867453746252064770029)),
396 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.462590910138598083940803704521211569234e-6)),
397 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -0.1735307814426389420248044907765671743012e-9)),
398 };
399 T result = 0;
400 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
401 {
402 result += (-d[k-1]*dz)/(k*dz + k*k);
403 }
404 return result;
405 }
406
407 template<class T>
408 static T lanczos_sum_near_2(const T& dz)
409 {
410 lanczos_initializer<lanczos13, T>::force_instantiate(); // Ensure our constants get initialized before main()
411 static const T d[12] = {
412 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 26.96979819614830698367887026728396466395)),
413 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -110.8705424709385114023884328797900204863)),
414 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 189.7258846119231466417015694690434770085)),
415 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -175.3397202971107486383321670769397356553)),
416 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 95.03437648691551457087250340903980824948)),
417 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -30.7406022781665264273675797983497141978)),
418 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 5.792405601630517993355102578874590410552)),
419 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -0.5951993240669148697377539518639997795831)),
420 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.02944979359164017509944724739946255067671)),
421 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -0.0005586586555377030921194246330399163602684)),
422 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, 0.2581888478270733025288922038673392636029e-5)),
423 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 72, -0.9685385411006641478305219367315965391289e-9)),
424 };
425 T result = 0;
426 T z = dz + 2;
427 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
428 {
429 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
430 }
431 return result;
432 }
433
434 static double g(){ return 13.1445650000000000545696821063756942749; }
435 };
436
437 //
438 // Lanczos Coefficients for N=22 G=22.61891
439 // Max experimental error (with arbitary precision arithmetic) 2.9524e-38
440 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
441 //
442 struct lanczos22 : public mpl::int_<120>
443 {
444 //
445 // Produces slightly better than 128-bit long-double precision when
446 // evaluated at higher precision:
447 //
448 template <class T>
449 static T lanczos_sum(const T& z)
450 {
451 lanczos_initializer<lanczos22, T>::force_instantiate(); // Ensure our constants get initialized before main()
452 static const T num[22] = {
453 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 46198410803245094237463011094.12173081986)),
454 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 43735859291852324413622037436.321513777)),
455 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 19716607234435171720534556386.97481377748)),
456 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 5629401471315018442177955161.245623932129)),
457 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 1142024910634417138386281569.245580222392)),
458 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 175048529315951173131586747.695329230778)),
459 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 21044290245653709191654675.41581372963167)),
460 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 2033001410561031998451380.335553678782601)),
461 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 160394318862140953773928.8736211601848891)),
462 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 10444944438396359705707.48957290388740896)),
463 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 565075825801617290121.1466393747967538948)),
464 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 25475874292116227538.99448534450411942597)),
465 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 957135055846602154.6720835535232270205725)),
466 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 29874506304047462.23662392445173880821515)),
467 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 769651310384737.2749087590725764959689181)),
468 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 16193289100889.15989633624378404096011797)),
469 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 273781151680.6807433264462376754578933261)),
470 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 3630485900.32917021712188739762161583295)),
471 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 36374352.05577334277856865691538582936484)),
472 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 258945.7742115532455441786924971194951043)),
473 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 1167.501919472435718934219997431551246996)),
474 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 2.50662827463100050241576528481104525333))
475 };
476 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint64_t) denom[22] = {
477 BOOST_MATH_INT_VALUE_SUFFIX(0, uLL),
478 BOOST_MATH_INT_VALUE_SUFFIX(2432902008176640000, uLL),
479 BOOST_MATH_INT_VALUE_SUFFIX(8752948036761600000, uLL),
480 BOOST_MATH_INT_VALUE_SUFFIX(13803759753640704000, uLL),
481 BOOST_MATH_INT_VALUE_SUFFIX(12870931245150988800, uLL),
482 BOOST_MATH_INT_VALUE_SUFFIX(8037811822645051776, uLL),
483 BOOST_MATH_INT_VALUE_SUFFIX(3599979517947607200, uLL),
484 BOOST_MATH_INT_VALUE_SUFFIX(1206647803780373360, uLL),
485 BOOST_MATH_INT_VALUE_SUFFIX(311333643161390640, uLL),
486 BOOST_MATH_INT_VALUE_SUFFIX(63030812099294896, uLL),
487 BOOST_MATH_INT_VALUE_SUFFIX(10142299865511450, uLL),
488 BOOST_MATH_INT_VALUE_SUFFIX(1307535010540395, uLL),
489 BOOST_MATH_INT_VALUE_SUFFIX(135585182899530, uLL),
490 BOOST_MATH_INT_VALUE_SUFFIX(11310276995381, uLL),
491 BOOST_MATH_INT_VALUE_SUFFIX(756111184500, uLL),
492 BOOST_MATH_INT_VALUE_SUFFIX(40171771630, uLL),
493 BOOST_MATH_INT_VALUE_SUFFIX(1672280820, uLL),
494 BOOST_MATH_INT_VALUE_SUFFIX(53327946, uLL),
495 BOOST_MATH_INT_VALUE_SUFFIX(1256850, uLL),
496 BOOST_MATH_INT_VALUE_SUFFIX(20615, uLL),
497 BOOST_MATH_INT_VALUE_SUFFIX(210, uLL),
498 BOOST_MATH_INT_VALUE_SUFFIX(1, uLL)
499 };
500 return boost::math::tools::evaluate_rational(num, denom, z);
501 }
502
503 template <class T>
504 static T lanczos_sum_expG_scaled(const T& z)
505 {
506 lanczos_initializer<lanczos22, T>::force_instantiate(); // Ensure our constants get initialized before main()
507 static const T num[22] = {
508 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 6939996264376682180.277485395074954356211)),
509 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 6570067992110214451.87201438870245659384)),
510 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 2961859037444440551.986724631496417064121)),
511 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 845657339772791245.3541226499766163431651)),
512 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 171556737035449095.2475716923888737881837)),
513 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 26296059072490867.7822441885603400926007)),
514 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 3161305619652108.433798300149816829198706)),
515 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 305400596026022.4774396904484542582526472)),
516 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 24094681058862.55120507202622377623528108)),
517 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 1569055604375.919477574824168939428328839)),
518 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 84886558909.02047889339710230696942513159)),
519 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 3827024985.166751989686050643579753162298)),
520 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 143782298.9273215199098728674282885500522)),
521 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 4487794.24541641841336786238909171265944)),
522 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 115618.2025760830513505888216285273541959)),
523 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 2432.580773108508276957461757328744780439)),
524 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 41.12782532742893597168530008461874360191)),
525 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.5453771709477689805460179187388702295792)),
526 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.005464211062612080347167337964166505282809)),
527 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.388992321263586767037090706042788910953e-4)),
528 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.1753839324538447655939518484052327068859e-6)),
529 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.3765495513732730583386223384116545391759e-9))
530 };
531 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint64_t) denom[22] = {
532 BOOST_MATH_INT_VALUE_SUFFIX(0, uLL),
533 BOOST_MATH_INT_VALUE_SUFFIX(2432902008176640000, uLL),
534 BOOST_MATH_INT_VALUE_SUFFIX(8752948036761600000, uLL),
535 BOOST_MATH_INT_VALUE_SUFFIX(13803759753640704000, uLL),
536 BOOST_MATH_INT_VALUE_SUFFIX(12870931245150988800, uLL),
537 BOOST_MATH_INT_VALUE_SUFFIX(8037811822645051776, uLL),
538 BOOST_MATH_INT_VALUE_SUFFIX(3599979517947607200, uLL),
539 BOOST_MATH_INT_VALUE_SUFFIX(1206647803780373360, uLL),
540 BOOST_MATH_INT_VALUE_SUFFIX(311333643161390640, uLL),
541 BOOST_MATH_INT_VALUE_SUFFIX(63030812099294896, uLL),
542 BOOST_MATH_INT_VALUE_SUFFIX(10142299865511450, uLL),
543 BOOST_MATH_INT_VALUE_SUFFIX(1307535010540395, uLL),
544 BOOST_MATH_INT_VALUE_SUFFIX(135585182899530, uLL),
545 BOOST_MATH_INT_VALUE_SUFFIX(11310276995381, uLL),
546 BOOST_MATH_INT_VALUE_SUFFIX(756111184500, uLL),
547 BOOST_MATH_INT_VALUE_SUFFIX(40171771630, uLL),
548 BOOST_MATH_INT_VALUE_SUFFIX(1672280820, uLL),
549 BOOST_MATH_INT_VALUE_SUFFIX(53327946, uLL),
550 BOOST_MATH_INT_VALUE_SUFFIX(1256850, uLL),
551 BOOST_MATH_INT_VALUE_SUFFIX(20615, uLL),
552 BOOST_MATH_INT_VALUE_SUFFIX(210, uLL),
553 BOOST_MATH_INT_VALUE_SUFFIX(1, uLL)
554 };
555 return boost::math::tools::evaluate_rational(num, denom, z);
556 }
557
558
559 template<class T>
560 static T lanczos_sum_near_1(const T& dz)
561 {
562 lanczos_initializer<lanczos22, T>::force_instantiate(); // Ensure our constants get initialized before main()
563 static const T d[21] = {
564 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 8.318998691953337183034781139546384476554)),
565 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -63.15415991415959158214140353299240638675)),
566 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 217.3108224383632868591462242669081540163)),
567 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -448.5134281386108366899784093610397354889)),
568 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 619.2903759363285456927248474593012711346)),
569 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -604.1630177420625418522025080080444177046)),
570 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 428.8166750424646119935047118287362193314)),
571 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -224.6988753721310913866347429589434550302)),
572 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 87.32181627555510833499451817622786940961)),
573 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -25.07866854821128965662498003029199058098)),
574 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 5.264398125689025351448861011657789005392)),
575 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.792518936256495243383586076579921559914)),
576 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.08317448364744713773350272460937904691566)),
577 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.005845345166274053157781068150827567998882)),
578 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.0002599412126352082483326238522490030412391)),
579 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.6748102079670763884917431338234783496303e-5)),
580 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.908824383434109002762325095643458603605e-7)),
581 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.5299325929309389890892469299969669579725e-9)),
582 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.994306085859549890267983602248532869362e-12)),
583 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.3499893692975262747371544905820891835298e-15)),
584 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.7260746353663365145454867069182884694961e-20)),
585 };
586 T result = 0;
587 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
588 {
589 result += (-d[k-1]*dz)/(k*dz + k*k);
590 }
591 return result;
592 }
593
594 template<class T>
595 static T lanczos_sum_near_2(const T& dz)
596 {
597 lanczos_initializer<lanczos22, T>::force_instantiate(); // Ensure our constants get initialized before main()
598 static const T d[21] = {
599 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 75.39272007105208086018421070699575462226)),
600 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -572.3481967049935412452681346759966390319)),
601 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 1969.426202741555335078065370698955484358)),
602 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -4064.74968778032030891520063865996757519)),
603 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 5612.452614138013929794736248384309574814)),
604 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -5475.357667500026172903620177988213902339)),
605 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 3886.243614216111328329547926490398103492)),
606 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -2036.382026072125407192448069428134470564)),
607 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 791.3727954936062108045551843636692287652)),
608 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -227.2808432388436552794021219198885223122)),
609 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 47.70974355562144229897637024320739257284)),
610 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -7.182373807798293545187073539819697141572)),
611 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.7537866989631514559601547530490976100468)),
612 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.05297470142240154822658739758236594717787)),
613 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.00235577330936380542539812701472320434133)),
614 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.6115613067659273118098229498679502138802e-4)),
615 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.8236417010170941915758315020695551724181e-6)),
616 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.4802628430993048190311242611330072198089e-8)),
617 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.9011113376981524418952720279739624707342e-11)),
618 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, -0.3171854152689711198382455703658589996796e-14)),
619 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 120, 0.6580207998808093935798753964580596673177e-19)),
620 };
621 T result = 0;
622 T z = dz + 2;
623 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
624 {
625 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
626 }
627 return result;
628 }
629
630 static double g(){ return 22.61890999999999962710717227309942245483; }
631 };
632
633 //
634 // Lanczos Coefficients for N=6 G=1.428456135094165802001953125
635 // Max experimental error (with arbitary precision arithmetic) 8.111667e-8
636 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
637 //
638 struct lanczos6m24 : public mpl::int_<24>
639 {
640 //
641 // Use for float precision, when evaluated as a float:
642 //
643 template <class T>
644 static T lanczos_sum(const T& z)
645 {
646 static const T num[6] = {
647 static_cast<T>(58.52061591769095910314047740215847630266L),
648 static_cast<T>(182.5248962595894264831189414768236280862L),
649 static_cast<T>(211.0971093028510041839168287718170827259L),
650 static_cast<T>(112.2526547883668146736465390902227161763L),
651 static_cast<T>(27.5192015197455403062503721613097825345L),
652 static_cast<T>(2.50662858515256974113978724717473206342L)
653 };
654 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint16_t) denom[6] = {
655 static_cast<boost::uint16_t>(0u),
656 static_cast<boost::uint16_t>(24u),
657 static_cast<boost::uint16_t>(50u),
658 static_cast<boost::uint16_t>(35u),
659 static_cast<boost::uint16_t>(10u),
660 static_cast<boost::uint16_t>(1u)
661 };
662 return boost::math::tools::evaluate_rational(num, denom, z);
663 }
664
665 template <class T>
666 static T lanczos_sum_expG_scaled(const T& z)
667 {
668 static const T num[6] = {
669 static_cast<T>(14.0261432874996476619570577285003839357L),
670 static_cast<T>(43.74732405540314316089531289293124360129L),
671 static_cast<T>(50.59547402616588964511581430025589038612L),
672 static_cast<T>(26.90456680562548195593733429204228910299L),
673 static_cast<T>(6.595765571169314946316366571954421695196L),
674 static_cast<T>(0.6007854010515290065101128585795542383721L)
675 };
676 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint16_t) denom[6] = {
677 static_cast<boost::uint16_t>(0u),
678 static_cast<boost::uint16_t>(24u),
679 static_cast<boost::uint16_t>(50u),
680 static_cast<boost::uint16_t>(35u),
681 static_cast<boost::uint16_t>(10u),
682 static_cast<boost::uint16_t>(1u)
683 };
684 return boost::math::tools::evaluate_rational(num, denom, z);
685 }
686
687
688 template<class T>
689 static T lanczos_sum_near_1(const T& dz)
690 {
691 static const T d[5] = {
692 static_cast<T>(0.4922488055204602807654354732674868442106L),
693 static_cast<T>(0.004954497451132152436631238060933905650346L),
694 static_cast<T>(-0.003374784572167105840686977985330859371848L),
695 static_cast<T>(0.001924276018962061937026396537786414831385L),
696 static_cast<T>(-0.00056533046336427583708166383712907694434L),
697 };
698 T result = 0;
699 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
700 {
701 result += (-d[k-1]*dz)/(k*dz + k*k);
702 }
703 return result;
704 }
705
706 template<class T>
707 static T lanczos_sum_near_2(const T& dz)
708 {
709 static const T d[5] = {
710 static_cast<T>(0.6534966888520080645505805298901130485464L),
711 static_cast<T>(0.006577461728560758362509168026049182707101L),
712 static_cast<T>(-0.004480276069269967207178373559014835978161L),
713 static_cast<T>(0.00255461870648818292376982818026706528842L),
714 static_cast<T>(-0.000750517993690428370380996157470900204524L),
715 };
716 T result = 0;
717 T z = dz + 2;
718 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
719 {
720 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
721 }
722 return result;
723 }
724
725 static double g(){ return 1.428456135094165802001953125; }
726 };
727
728 //
729 // Lanczos Coefficients for N=13 G=6.024680040776729583740234375
730 // Max experimental error (with arbitary precision arithmetic) 1.196214e-17
731 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
732 //
733 struct lanczos13m53 : public mpl::int_<53>
734 {
735 //
736 // Use for double precision, when evaluated as a double:
737 //
738 template <class T>
739 static T lanczos_sum(const T& z)
740 {
741 static const T num[13] = {
742 static_cast<T>(23531376880.41075968857200767445163675473L),
743 static_cast<T>(42919803642.64909876895789904700198885093L),
744 static_cast<T>(35711959237.35566804944018545154716670596L),
745 static_cast<T>(17921034426.03720969991975575445893111267L),
746 static_cast<T>(6039542586.35202800506429164430729792107L),
747 static_cast<T>(1439720407.311721673663223072794912393972L),
748 static_cast<T>(248874557.8620541565114603864132294232163L),
749 static_cast<T>(31426415.58540019438061423162831820536287L),
750 static_cast<T>(2876370.628935372441225409051620849613599L),
751 static_cast<T>(186056.2653952234950402949897160456992822L),
752 static_cast<T>(8071.672002365816210638002902272250613822L),
753 static_cast<T>(210.8242777515793458725097339207133627117L),
754 static_cast<T>(2.506628274631000270164908177133837338626L)
755 };
756 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint32_t) denom[13] = {
757 static_cast<boost::uint32_t>(0u),
758 static_cast<boost::uint32_t>(39916800u),
759 static_cast<boost::uint32_t>(120543840u),
760 static_cast<boost::uint32_t>(150917976u),
761 static_cast<boost::uint32_t>(105258076u),
762 static_cast<boost::uint32_t>(45995730u),
763 static_cast<boost::uint32_t>(13339535u),
764 static_cast<boost::uint32_t>(2637558u),
765 static_cast<boost::uint32_t>(357423u),
766 static_cast<boost::uint32_t>(32670u),
767 static_cast<boost::uint32_t>(1925u),
768 static_cast<boost::uint32_t>(66u),
769 static_cast<boost::uint32_t>(1u)
770 };
771 return boost::math::tools::evaluate_rational(num, denom, z);
772 }
773
774 template <class T>
775 static T lanczos_sum_expG_scaled(const T& z)
776 {
777 static const T num[13] = {
778 static_cast<T>(56906521.91347156388090791033559122686859L),
779 static_cast<T>(103794043.1163445451906271053616070238554L),
780 static_cast<T>(86363131.28813859145546927288977868422342L),
781 static_cast<T>(43338889.32467613834773723740590533316085L),
782 static_cast<T>(14605578.08768506808414169982791359218571L),
783 static_cast<T>(3481712.15498064590882071018964774556468L),
784 static_cast<T>(601859.6171681098786670226533699352302507L),
785 static_cast<T>(75999.29304014542649875303443598909137092L),
786 static_cast<T>(6955.999602515376140356310115515198987526L),
787 static_cast<T>(449.9445569063168119446858607650988409623L),
788 static_cast<T>(19.51992788247617482847860966235652136208L),
789 static_cast<T>(0.5098416655656676188125178644804694509993L),
790 static_cast<T>(0.006061842346248906525783753964555936883222L)
791 };
792 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint32_t) denom[13] = {
793 static_cast<boost::uint32_t>(0u),
794 static_cast<boost::uint32_t>(39916800u),
795 static_cast<boost::uint32_t>(120543840u),
796 static_cast<boost::uint32_t>(150917976u),
797 static_cast<boost::uint32_t>(105258076u),
798 static_cast<boost::uint32_t>(45995730u),
799 static_cast<boost::uint32_t>(13339535u),
800 static_cast<boost::uint32_t>(2637558u),
801 static_cast<boost::uint32_t>(357423u),
802 static_cast<boost::uint32_t>(32670u),
803 static_cast<boost::uint32_t>(1925u),
804 static_cast<boost::uint32_t>(66u),
805 static_cast<boost::uint32_t>(1u)
806 };
807 return boost::math::tools::evaluate_rational(num, denom, z);
808 }
809
810
811 template<class T>
812 static T lanczos_sum_near_1(const T& dz)
813 {
814 static const T d[12] = {
815 static_cast<T>(2.208709979316623790862569924861841433016L),
816 static_cast<T>(-3.327150580651624233553677113928873034916L),
817 static_cast<T>(1.483082862367253753040442933770164111678L),
818 static_cast<T>(-0.1993758927614728757314233026257810172008L),
819 static_cast<T>(0.004785200610085071473880915854204301886437L),
820 static_cast<T>(-0.1515973019871092388943437623825208095123e-5L),
821 static_cast<T>(-0.2752907702903126466004207345038327818713e-7L),
822 static_cast<T>(0.3075580174791348492737947340039992829546e-7L),
823 static_cast<T>(-0.1933117898880828348692541394841204288047e-7L),
824 static_cast<T>(0.8690926181038057039526127422002498960172e-8L),
825 static_cast<T>(-0.2499505151487868335680273909354071938387e-8L),
826 static_cast<T>(0.3394643171893132535170101292240837927725e-9L),
827 };
828 T result = 0;
829 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
830 {
831 result += (-d[k-1]*dz)/(k*dz + k*k);
832 }
833 return result;
834 }
835
836 template<class T>
837 static T lanczos_sum_near_2(const T& dz)
838 {
839 static const T d[12] = {
840 static_cast<T>(6.565936202082889535528455955485877361223L),
841 static_cast<T>(-9.8907772644920670589288081640128194231L),
842 static_cast<T>(4.408830289125943377923077727900630927902L),
843 static_cast<T>(-0.5926941084905061794445733628891024027949L),
844 static_cast<T>(0.01422519127192419234315002746252160965831L),
845 static_cast<T>(-0.4506604409707170077136555010018549819192e-5L),
846 static_cast<T>(-0.8183698410724358930823737982119474130069e-7L),
847 static_cast<T>(0.9142922068165324132060550591210267992072e-7L),
848 static_cast<T>(-0.5746670642147041587497159649318454348117e-7L),
849 static_cast<T>(0.2583592566524439230844378948704262291927e-7L),
850 static_cast<T>(-0.7430396708998719707642735577238449585822e-8L),
851 static_cast<T>(0.1009141566987569892221439918230042368112e-8L),
852 };
853 T result = 0;
854 T z = dz + 2;
855 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
856 {
857 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
858 }
859 return result;
860 }
861
862 static double g(){ return 6.024680040776729583740234375; }
863 };
864
865 //
866 // Lanczos Coefficients for N=17 G=12.2252227365970611572265625
867 // Max experimental error (with arbitary precision arithmetic) 2.7699e-26
868 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
869 //
870 struct lanczos17m64 : public mpl::int_<64>
871 {
872 //
873 // Use for extended-double precision, when evaluated as an extended-double:
874 //
875 template <class T>
876 static T lanczos_sum(const T& z)
877 {
878 lanczos_initializer<lanczos17m64, T>::force_instantiate(); // Ensure our constants get initialized before main()
879 static const T num[17] = {
880 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 553681095419291969.2230556393350368550504)),
881 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 731918863887667017.2511276782146694632234)),
882 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 453393234285807339.4627124634539085143364)),
883 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 174701893724452790.3546219631779712198035)),
884 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 46866125995234723.82897281620357050883077)),
885 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9281280675933215.169109622777099699054272)),
886 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1403600894156674.551057997617468721789536)),
887 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 165345984157572.7305349809894046783973837)),
888 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 15333629842677.31531822808737907246817024)),
889 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1123152927963.956626161137169462874517318)),
890 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 64763127437.92329018717775593533620578237)),
891 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2908830362.657527782848828237106640944457)),
892 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 99764700.56999856729959383751710026787811)),
893 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2525791.604886139959837791244686290089331)),
894 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 44516.94034970167828580039370201346554872)),
895 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 488.0063567520005730476791712814838113252)),
896 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2.50662827463100050241576877135758834683))
897 };
898 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint64_t) denom[17] = {
899 BOOST_MATH_INT_VALUE_SUFFIX(0, uLL),
900 BOOST_MATH_INT_VALUE_SUFFIX(1307674368000, uLL),
901 BOOST_MATH_INT_VALUE_SUFFIX(4339163001600, uLL),
902 BOOST_MATH_INT_VALUE_SUFFIX(6165817614720, uLL),
903 BOOST_MATH_INT_VALUE_SUFFIX(5056995703824, uLL),
904 BOOST_MATH_INT_VALUE_SUFFIX(2706813345600, uLL),
905 BOOST_MATH_INT_VALUE_SUFFIX(1009672107080, uLL),
906 BOOST_MATH_INT_VALUE_SUFFIX(272803210680, uLL),
907 BOOST_MATH_INT_VALUE_SUFFIX(54631129553, uLL),
908 BOOST_MATH_INT_VALUE_SUFFIX(8207628000, uLL),
909 BOOST_MATH_INT_VALUE_SUFFIX(928095740, uLL),
910 BOOST_MATH_INT_VALUE_SUFFIX(78558480, uLL),
911 BOOST_MATH_INT_VALUE_SUFFIX(4899622, uLL),
912 BOOST_MATH_INT_VALUE_SUFFIX(218400, uLL),
913 BOOST_MATH_INT_VALUE_SUFFIX(6580, uLL),
914 BOOST_MATH_INT_VALUE_SUFFIX(120, uLL),
915 BOOST_MATH_INT_VALUE_SUFFIX(1, uLL)
916 };
917 return boost::math::tools::evaluate_rational(num, denom, z);
918 }
919
920 template <class T>
921 static T lanczos_sum_expG_scaled(const T& z)
922 {
923 lanczos_initializer<lanczos17m64, T>::force_instantiate(); // Ensure our constants get initialized before main()
924 static const T num[17] = {
925 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2715894658327.717377557655133124376674911)),
926 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 3590179526097.912105038525528721129550434)),
927 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 2223966599737.814969312127353235818710172)),
928 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 856940834518.9562481809925866825485883417)),
929 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 229885871668.749072933597446453399395469)),
930 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 45526171687.54610815813502794395753410032)),
931 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 6884887713.165178784550917647709216424823)),
932 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 811048596.1407531864760282453852372777439)),
933 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 75213915.96540822314499613623119501704812)),
934 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 5509245.417224265151697527957954952830126)),
935 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 317673.5368435419126714931842182369574221)),
936 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 14268.27989845035520147014373320337523596)),
937 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 489.3618720403263670213909083601787814792)),
938 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 12.38941330038454449295883217865458609584)),
939 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.2183627389504614963941574507281683147897)),
940 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.002393749522058449186690627996063983095463)),
941 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1229541408909435212800785616808830746135e-4))
942 };
943 static const BOOST_MATH_INT_TABLE_TYPE(T, boost::uint64_t) denom[17] = {
944 BOOST_MATH_INT_VALUE_SUFFIX(0, uLL),
945 BOOST_MATH_INT_VALUE_SUFFIX(1307674368000, uLL),
946 BOOST_MATH_INT_VALUE_SUFFIX(4339163001600, uLL),
947 BOOST_MATH_INT_VALUE_SUFFIX(6165817614720, uLL),
948 BOOST_MATH_INT_VALUE_SUFFIX(5056995703824, uLL),
949 BOOST_MATH_INT_VALUE_SUFFIX(2706813345600, uLL),
950 BOOST_MATH_INT_VALUE_SUFFIX(1009672107080, uLL),
951 BOOST_MATH_INT_VALUE_SUFFIX(272803210680, uLL),
952 BOOST_MATH_INT_VALUE_SUFFIX(54631129553, uLL),
953 BOOST_MATH_INT_VALUE_SUFFIX(8207628000, uLL),
954 BOOST_MATH_INT_VALUE_SUFFIX(928095740, uLL),
955 BOOST_MATH_INT_VALUE_SUFFIX(78558480, uLL),
956 BOOST_MATH_INT_VALUE_SUFFIX(4899622, uLL),
957 BOOST_MATH_INT_VALUE_SUFFIX(218400, uLL),
958 BOOST_MATH_INT_VALUE_SUFFIX(6580, uLL),
959 BOOST_MATH_INT_VALUE_SUFFIX(120, uLL),
960 BOOST_MATH_INT_VALUE_SUFFIX(1, uLL)
961 };
962 return boost::math::tools::evaluate_rational(num, denom, z);
963 }
964
965
966 template<class T>
967 static T lanczos_sum_near_1(const T& dz)
968 {
969 lanczos_initializer<lanczos17m64, T>::force_instantiate(); // Ensure our constants get initialized before main()
970 static const T d[16] = {
971 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 4.493645054286536365763334986866616581265)),
972 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -16.95716370392468543800733966378143997694)),
973 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 26.19196892983737527836811770970479846644)),
974 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -21.3659076437988814488356323758179283908)),
975 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 9.913992596774556590710751047594507535764)),
976 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -2.62888300018780199210536267080940382158)),
977 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.3807056693542503606384861890663080735588)),
978 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.02714647489697685807340312061034730486958)),
979 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.0007815484715461206757220527133967191796747)),
980 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.6108630817371501052576880554048972272435e-5)),
981 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.5037380238864836824167713635482801545086e-8)),
982 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.1483232144262638814568926925964858237006e-13)),
983 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1346609158752142460943888149156716841693e-14)),
984 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.660492688923978805315914918995410340796e-15)),
985 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.1472114697343266749193617793755763792681e-15)),
986 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.1410901942033374651613542904678399264447e-16)),
987 };
988 T result = 0;
989 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
990 {
991 result += (-d[k-1]*dz)/(k*dz + k*k);
992 }
993 return result;
994 }
995
996 template<class T>
997 static T lanczos_sum_near_2(const T& dz)
998 {
999 lanczos_initializer<lanczos17m64, T>::force_instantiate(); // Ensure our constants get initialized before main()
1000 static const T d[16] = {
1001 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 23.56409085052261327114594781581930373708)),
1002 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -88.92116338946308797946237246006238652361)),
1003 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 137.3472822086847596961177383569603988797)),
1004 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -112.0400438263562152489272966461114852861)),
1005 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 51.98768915202973863076166956576777843805)),
1006 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -13.78552090862799358221343319574970124948)),
1007 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 1.996371068830872830250406773917646121742)),
1008 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.1423525874909934506274738563671862576161)),
1009 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.004098338646046865122459664947239111298524)),
1010 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.3203286637326511000882086573060433529094e-4)),
1011 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.2641536751640138646146395939004587594407e-7)),
1012 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.7777876663062235617693516558976641009819e-13)),
1013 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.7061443477097101636871806229515157914789e-14)),
1014 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.3463537849537988455590834887691613484813e-14)),
1015 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, 0.7719578215795234036320348283011129450595e-15)),
1016 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 64, -0.7398586479708476329563577384044188912075e-16)),
1017 };
1018 T result = 0;
1019 T z = dz + 2;
1020 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
1021 {
1022 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
1023 }
1024 return result;
1025 }
1026
1027 static double g(){ return 12.2252227365970611572265625; }
1028 };
1029
1030 //
1031 // Lanczos Coefficients for N=24 G=20.3209821879863739013671875
1032 // Max experimental error (with arbitary precision arithmetic) 1.0541e-38
1033 // Generated with compiler: Microsoft Visual C++ version 8.0 on Win32 at Mar 23 2006
1034 //
1035 struct lanczos24m113 : public mpl::int_<113>
1036 {
1037 //
1038 // Use for long-double precision, when evaluated as an long-double:
1039 //
1040 template <class T>
1041 static T lanczos_sum(const T& z)
1042 {
1043 lanczos_initializer<lanczos24m113, T>::force_instantiate(); // Ensure our constants get initialized before main()
1044 static const T num[24] = {
1045 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2029889364934367661624137213253.22102954656825019111612712252027267955023987678816620961507)),
1046 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2338599599286656537526273232565.2727349714338768161421882478417543004440597874814359063158)),
1047 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1288527989493833400335117708406.3953711906175960449186720680201425446299360322830739180195)),
1048 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 451779745834728745064649902914.550539158066332484594436145043388809847364393288132164411521)),
1049 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 113141284461097964029239556815.291212318665536114012605167994061291631013303788706545334708)),
1050 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 21533689802794625866812941616.7509064680880468667055339259146063256555368135236149614592432)),
1051 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3235510315314840089932120340.71494940111731241353655381919722177496659303550321056514776757)),
1052 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 393537392344185475704891959.081297108513472083749083165179784098220158201055270548272414314)),
1053 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 39418265082950435024868801.5005452240816902251477336582325944930252142622315101857742955673)),
1054 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3290158764187118871697791.05850632319194734270969161036889516414516566453884272345518372696)),
1055 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 230677110449632078321772.618245845856640677845629174549731890660612368500786684333975350954)),
1056 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 13652233645509183190158.5916189185218250859402806777406323001463296297553612462737044693697)),
1057 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 683661466754325350495.216655026531202476397782296585200982429378069417193575896602446904762)),
1058 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 28967871782219334117.0122379171041074970463982134039409352925258212207710168851968215545064)),
1059 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1036104088560167006.2022834098572346459442601718514554488352117620272232373622553429728555)),
1060 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 31128490785613152.8380102669349814751268126141105475287632676569913936040772990253369753962)),
1061 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 779327504127342.536207878988196814811198475410572992436243686674896894543126229424358472541)),
1062 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 16067543181294.643350688789124777020407337133926174150582333950666044399234540521336771876)),
1063 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 268161795520.300916569439413185778557212729611517883948634711190170998896514639936969855484)),
1064 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3533216359.10528191668842486732408440112703691790824611391987708562111396961696753452085068)),
1065 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 35378979.5479656110614685178752543826919239614088343789329169535932709470588426584501652577)),
1066 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 253034.881362204346444503097491737872930637147096453940375713745904094735506180552724766444)),
1067 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1151.61895453463992438325318456328526085882924197763140514450975619271382783957699017875304)),
1068 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2.50662827463100050241576528481104515966515623051532908941425544355490413900497467936202516))
1069 };
1070 static const T denom[24] = {
1071 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0)),
1072 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.112400072777760768e22)),
1073 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.414847677933545472e22)),
1074 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 6756146673770930688000.0)),
1075 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 6548684852703068697600.0)),
1076 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4280722865357147142912.0)),
1077 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2021687376910682741568.0)),
1078 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 720308216440924653696.0)),
1079 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 199321978221066137360.0)),
1080 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 43714229649594412832.0)),
1081 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 7707401101297361068.0)),
1082 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1103230881185949736.0)),
1083 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 129006659818331295.0)),
1084 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 12363045847086207.0)),
1085 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 971250460939913.0)),
1086 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 62382416421941.0)),
1087 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3256091103430.0)),
1088 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 136717357942.0)),
1089 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4546047198.0)),
1090 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 116896626)),
1091 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2240315)),
1092 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 30107)),
1093 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 253)),
1094 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1))
1095 };
1096 return boost::math::tools::evaluate_rational(num, denom, z);
1097 }
1098
1099 template <class T>
1100 static T lanczos_sum_expG_scaled(const T& z)
1101 {
1102 lanczos_initializer<lanczos24m113, T>::force_instantiate(); // Ensure our constants get initialized before main()
1103 static const T num[24] = {
1104 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3035162425359883494754.02878223286972654682199012688209026810841953293372712802258398358538)),
1105 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3496756894406430103600.16057175075063458536101374170860226963245118484234495645518505519827)),
1106 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1926652656689320888654.01954015145958293168365236755537645929361841917596501251362171653478)),
1107 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 675517066488272766316.083023742440619929434602223726894748181327187670231286180156444871912)),
1108 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 169172853104918752780.086262749564831660238912144573032141700464995906149421555926000038492)),
1109 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 32197935167225605785.6444116302160245528783954573163541751756353183343357329404208062043808)),
1110 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4837849542714083249.37587447454818124327561966323276633775195138872820542242539845253171632)),
1111 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 588431038090493242.308438203986649553459461798968819276505178004064031201740043314534404158)),
1112 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 58939585141634058.6206417889192563007809470547755357240808035714047014324843817783741669733)),
1113 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4919561837722192.82991866530802080996138070630296720420704876654726991998309206256077395868)),
1114 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 344916580244240.407442753122831512004021081677987651622305356145640394384006997569631719101)),
1115 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 20413302960687.8250598845969238472629322716685686993835561234733641729957841485003560103066)),
1116 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1022234822943.78400752460970689311934727763870970686747383486600540378889311406851534545789)),
1117 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 43313787191.9821354846952908076307094286897439975815501673706144217246093900159173598852503)),
1118 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1549219505.59667418528481770869280437577581951167003505825834192510436144666564648361001914)),
1119 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 46544421.1998761919380541579358096705925369145324466147390364674998568485110045455014967149)),
1120 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1165278.06807504975090675074910052763026564833951579556132777702952882101173607903881127542)),
1121 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 24024.759267256769471083727721827405338569868270177779485912486668586611981795179894572115)),
1122 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 400.965008113421955824358063769761286758463521789765880962939528760888853281920872064838918)),
1123 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 5.28299015654478269617039029170846385138134929147421558771949982217659507918482272439717603)),
1124 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.0528999024412510102409256676599360516359062802002483877724963720047531347449011629466149805)),
1125 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.000378346710654740685454266569593414561162134092347356968516522170279688139165340746957511115)),
1126 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.172194142179211139195966608011235161516824700287310869949928393345257114743230967204370963e-5)),
1127 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.374799931707148855771381263542708435935402853962736029347951399323367765509988401336565436e-8))
1128 };
1129 static const T denom[24] = {
1130 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0)),
1131 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.112400072777760768e22)),
1132 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.414847677933545472e22)),
1133 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 6756146673770930688000.0)),
1134 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 6548684852703068697600.0)),
1135 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4280722865357147142912.0)),
1136 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2021687376910682741568.0)),
1137 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 720308216440924653696.0)),
1138 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 199321978221066137360.0)),
1139 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 43714229649594412832.0)),
1140 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 7707401101297361068.0)),
1141 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1103230881185949736.0)),
1142 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 129006659818331295.0)),
1143 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 12363045847086207.0)),
1144 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 971250460939913.0)),
1145 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 62382416421941.0)),
1146 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 3256091103430.0)),
1147 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 136717357942.0)),
1148 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4546047198.0)),
1149 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 116896626)),
1150 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2240315)),
1151 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 30107)),
1152 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 253)),
1153 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1))
1154 };
1155 return boost::math::tools::evaluate_rational(num, denom, z);
1156 }
1157
1158
1159 template<class T>
1160 static T lanczos_sum_near_1(const T& dz)
1161 {
1162 lanczos_initializer<lanczos24m113, T>::force_instantiate(); // Ensure our constants get initialized before main()
1163 static const T d[23] = {
1164 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 7.4734083002469026177867421609938203388868806387315406134072298925733950040583068760685908)),
1165 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -50.4225805042247530267317342133388132970816607563062253708655085754357843064134941138154171)),
1166 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 152.288200621747008570784082624444625293884063492396162110698238568311211546361189979357019)),
1167 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -271.894959539150384169327513139846971255640842175739337449692360299099322742181325023644769)),
1168 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 319.240102980202312307047586791116902719088581839891008532114107693294261542869734803906793)),
1169 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -259.493144143048088289689500935518073716201741349569864988870534417890269467336454358361499)),
1170 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 149.747518319689708813209645403067832020714660918583227716408482877303972685262557460145835)),
1171 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -61.9261301009341333289187201425188698128684426428003249782448828881580630606817104372760037)),
1172 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 18.3077524177286961563937379403377462608113523887554047531153187277072451294845795496072365)),
1173 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -3.82011322251948043097070160584761236869363471824695092089556195047949392738162970152230254)),
1174 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.549382685505691522516705902336780999493262538301283190963770663549981309645795228539620711)),
1175 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.0524814679715180697633723771076668718265358076235229045603747927518423453658004287459638024)),
1176 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.00315392664003333528534120626687784812050217700942910879712808180705014754163256855643360698)),
1177 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.000110098373127648510519799564665442121339511198561008748083409549601095293123407080388658329)),
1178 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.19809382866681658224945717689377373458866950897791116315219376038432014207446832310901893e-5)),
1179 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.152278977408600291408265615203504153130482270424202400677280558181047344681214058227949755e-7)),
1180 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.364344768076106268872239259083188037615571711218395765792787047015406264051536972018235217e-10)),
1181 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.148897510480440424971521542520683536298361220674662555578951242811522959610991621951203526e-13)),
1182 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.261199241161582662426512749820666625442516059622425213340053324061794752786482115387573582e-18)),
1183 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.780072664167099103420998436901014795601783313858454665485256897090476089641613851903791529e-24)),
1184 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.303465867587106629530056603454807425512962762653755513440561256044986695349304176849392735e-24)),
1185 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.615420597971283870342083342286977366161772327800327789325710571275345878439656918541092056e-25)),
1186 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.499641233843540749369110053005439398774706583601830828776209650445427083113181961630763702e-26)),
1187 };
1188 T result = 0;
1189 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
1190 {
1191 result += (-d[k-1]*dz)/(k*dz + k*k);
1192 }
1193 return result;
1194 }
1195
1196 template<class T>
1197 static T lanczos_sum_near_2(const T& dz)
1198 {
1199 lanczos_initializer<lanczos24m113, T>::force_instantiate(); // Ensure our constants get initialized before main()
1200 static const T d[23] = {
1201 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 61.4165001061101455341808888883960361969557848005400286332291451422461117307237198559485365)),
1202 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -414.372973678657049667308134761613915623353625332248315105320470271523320700386200587519147)),
1203 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1251.50505818554680171298972755376376836161706773644771875668053742215217922228357204561873)),
1204 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -2234.43389421602399514176336175766511311493214354568097811220122848998413358085613880612158)),
1205 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 2623.51647746991904821899989145639147785427273427135380151752779100215839537090464785708684)),
1206 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -2132.51572435428751962745870184529534443305617818870214348386131243463614597272260797772423)),
1207 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 1230.62572059218405766499842067263311220019173335523810725664442147670956427061920234820189)),
1208 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -508.90919151163744999377586956023909888833335885805154492270846381061182696305011395981929)),
1209 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 150.453184562246579758706538566480316921938628645961177699894388251635886834047343195475395)),
1210 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -31.3937061525822497422230490071156186113405446381476081565548185848237169870395131828731397)),
1211 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 4.51482916590287954234936829724231512565732528859217337795452389161322923867318809206313688)),
1212 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.431292919341108177524462194102701868233551186625103849565527515201492276412231365776131952)),
1213 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.0259189820815586225636729971503340447445001375909094681698918294680345547092233915092128323)),
1214 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.000904788882557558697594884691337532557729219389814315972435534723829065673966567231504429712)),
1215 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.162793589759218213439218473348810982422449144393340433592232065020562974405674317564164312e-4)),
1216 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.125142926178202562426432039899709511761368233479483128438847484617555752948755923647214487e-6)),
1217 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.299418680048132583204152682950097239197934281178261879500770485862852229898797687301941982e-9)),
1218 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.122364035267809278675627784883078206654408225276233049012165202996967011873995261617995421e-12)),
1219 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.21465364366598631597052073538883430194257709353929022544344097235100199405814005393447785e-17)),
1220 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.641064035802907518396608051803921688237330857546406669209280666066685733941549058513986818e-23)),
1221 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.249388374622173329690271566855185869111237201309011956145463506483151054813346819490278951e-23)),
1222 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, -0.505752900177513489906064295001851463338022055787536494321532352380960774349054239257683149e-24)),
1223 static_cast<T>(BOOST_MATH_BIG_CONSTANT(T, 113, 0.410605371184590959139968810080063542546949719163227555918846829816144878123034347778284006e-25)),
1224 };
1225 T result = 0;
1226 T z = dz + 2;
1227 for(unsigned k = 1; k <= sizeof(d)/sizeof(d[0]); ++k)
1228 {
1229 result += (-d[k-1]*dz)/(z + k*z + k*k - 1);
1230 }
1231 return result;
1232 }
1233
1234 static double g(){ return 20.3209821879863739013671875; }
1235 };
1236
1237
1238 //
1239 // placeholder for no lanczos info available:
1240 //
1241 struct undefined_lanczos : public mpl::int_<INT_MAX - 1> { };
1242
1243 #if 0
1244 #ifndef BOOST_NO_LIMITS_COMPILE_TIME_CONSTANTS
1245 #define BOOST_MATH_FLT_DIGITS ::std::numeric_limits<float>::digits
1246 #define BOOST_MATH_DBL_DIGITS ::std::numeric_limits<double>::digits
1247 #define BOOST_MATH_LDBL_DIGITS ::std::numeric_limits<long double>::digits
1248 #else
1249 #define BOOST_MATH_FLT_DIGITS FLT_MANT_DIG
1250 #define BOOST_MATH_DBL_DIGITS DBL_MANT_DIG
1251 #define BOOST_MATH_LDBL_DIGITS LDBL_MANT_DIG
1252 #endif
1253 #endif
1254
1255 typedef mpl::list<
1256 lanczos6m24,
1257 /* lanczos6, */
1258 lanczos13m53,
1259 /* lanczos13, */
1260 lanczos17m64,
1261 lanczos24m113,
1262 lanczos22,
1263 undefined_lanczos> lanczos_list;
1264
1265 template <class Real, class Policy>
1266 struct lanczos
1267 {
1268 typedef typename mpl::if_<
1269 typename mpl::less_equal<
1270 typename policies::precision<Real, Policy>::type,
1271 mpl::int_<0>
1272 >::type,
1273 mpl::int_<INT_MAX - 2>,
1274 typename policies::precision<Real, Policy>::type
1275 >::type target_precision;
1276
1277 typedef typename mpl::deref<typename mpl::find_if<
1278 lanczos_list,
1279 mpl::less_equal<target_precision, mpl::_1> >::type>::type type;
1280 };
1281
1282 } // namespace lanczos
1283 } // namespace math
1284 } // namespace boost
1285
1286 #if !defined(_CRAYC) && !defined(__CUDACC__) && (!defined(__GNUC__) || (__GNUC__ > 3) || ((__GNUC__ == 3) && (__GNUC_MINOR__ > 3)))
1287 #if (defined(_M_IX86_FP) && (_M_IX86_FP >= 2)) || defined(__SSE2__)
1288 #include <boost/math/special_functions/detail/lanczos_sse2.hpp>
1289 #endif
1290 #endif
1291
1292 #endif // BOOST_MATH_SPECIAL_FUNCTIONS_LANCZOS
1293
1294
1295
1296