1 /* specfunc/bessel_olver.c
2 *
3 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
4 *
5 * This program is free software; you can redistribute it and/or modify
6 * it under the terms of the GNU General Public License as published by
7 * the Free Software Foundation; either version 3 of the License, or (at
8 * your option) any later version.
9 *
10 * This program is distributed in the hope that it will be useful, but
11 * WITHOUT ANY WARRANTY; without even the implied warranty of
12 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 * General Public License for more details.
14 *
15 * You should have received a copy of the GNU General Public License
16 * along with this program; if not, write to the Free Software
17 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
18 */
19
20 /* Author: G. Jungman */
21
22 #include <config.h>
23 #include <gsl/gsl_math.h>
24 #include <gsl/gsl_errno.h>
25 #include <gsl/gsl_sf_airy.h>
26
27 #include "error.h"
28
29 #include "bessel.h"
30 #include "bessel_olver.h"
31
32 #include "chebyshev.h"
33 #include "cheb_eval.c"
34
35 /* fit for f(x) = zofmzeta((x+1)/2), 0 <= mzeta <= 1 */
36 static double zofmzeta_a_data[20] = {
37 2.9332563730829348990,
38 0.4896518224847036624,
39 0.0228637617355380860,
40 -0.0001715731377284693,
41 -0.0000105927538148751,
42 1.0595602530419e-6,
43 -4.68016051691e-8,
44 5.8310020e-12,
45 1.766537581e-10,
46 -1.45034640e-11,
47 4.357772e-13,
48 4.60971e-14,
49 -2.57571e-14,
50 2.26468e-14,
51 -2.22053e-14,
52 2.08593e-14,
53 -1.84454e-14,
54 1.50150e-14,
55 -1.06506e-14,
56 5.5375e-15
57 };
58 static cheb_series zofmzeta_a_cs = {
59 zofmzeta_a_data,
60 19,
61 -1,1,
62 8
63 };
64
65
66 /* fit for f(x) = zofmzeta((9x+11)/2), 1 <= mzeta <= 10 */
67 static double zofmzeta_b_data[30] = {
68 22.40725276466303489,
69 10.39808258825165581,
70 1.092050144486018425,
71 -0.071111274777921604,
72 0.008990125336059704,
73 -0.001201950338088875,
74 0.000106686807968315,
75 0.000017406491576830,
76 -0.000014946669657805,
77 6.189984487752e-6,
78 -2.049466715178e-6,
79 5.87189458020e-7,
80 -1.46077514157e-7,
81 2.9803936132e-8,
82 -3.817692108e-9,
83 -4.66980416e-10,
84 5.83860334e-10,
85 -2.78825299e-10,
86 1.01682688e-10,
87 -3.1209928e-11,
88 8.111122e-12,
89 -1.663986e-12,
90 1.81364e-13,
91 5.3414e-14,
92 -4.7234e-14,
93 2.1689e-14,
94 -7.815e-15,
95 2.371e-15,
96 -6.04e-16,
97 1.20e-16
98 };
99 static cheb_series zofmzeta_b_cs = {
100 zofmzeta_b_data,
101 29,
102 -1,1,
103 15
104 };
105
106
107 /* fit for f(x) = zofmzeta(mz(x))/mz(x)^(3/2),
108 * mz(x) = (2/(x+1))^(2/3) 10
109 * 10 <= mzeta <= Inf
110 */
111 static double zofmzeta_c_data[11] = {
112 1.3824761227122911500,
113 0.0244856101686774245,
114 -0.0000842866496282540,
115 1.4656076569771e-6,
116 -3.14874099476e-8,
117 7.561134833e-10,
118 -1.94531643e-11,
119 5.245878e-13,
120 -1.46380e-14,
121 4.192e-16,
122 -1.23e-17
123 };
124 static cheb_series zofmzeta_c_cs = {
125 zofmzeta_c_data,
126 10,
127 -1,1,
128 6
129 };
130
131
132 /* Invert [Abramowitz+Stegun, 9.3.39].
133 * Assumes minus_zeta >= 0.
134 */
135 double
gsl_sf_bessel_Olver_zofmzeta(double minus_zeta)136 gsl_sf_bessel_Olver_zofmzeta(double minus_zeta)
137 {
138 if(minus_zeta < 1.0) {
139 const double x = 2.0*minus_zeta - 1.0;
140 gsl_sf_result c;
141 cheb_eval_e(&zofmzeta_a_cs, x, &c);
142 return c.val;
143 }
144 else if(minus_zeta < 10.0) {
145 const double x = (2.0*minus_zeta - 11.0)/9.0;
146 gsl_sf_result c;
147 cheb_eval_e(&zofmzeta_b_cs, x, &c);
148 return c.val;
149 }
150 else {
151 const double TEN_32 = 31.62277660168379332; /* 10^(3/2) */
152 const double p = pow(minus_zeta, 3.0/2.0);
153 const double x = 2.0*TEN_32/p - 1.0;
154 gsl_sf_result c;
155 cheb_eval_e(&zofmzeta_c_cs, x, &c);
156 return c.val * p;
157 }
158 }
159
160
161 /* Chebyshev fit for f(x) = z(x)^6 A_3(z(x)), z(x) = 22/(10(x+1)) */
162 static double A3_gt1_data[31] = {
163 -0.123783199829515294670493131190,
164 0.104636462534700704670877382304,
165 -0.067500816575851826744877535903,
166 0.035563362418888483652711005520,
167 -0.0160738524035979408472979609051,
168 0.0064497878252851092073278056238,
169 -0.00235408261133449663958121821593,
170 0.00079545702851302155411892534965,
171 -0.00025214920745855079895784825637,
172 0.00007574004596069392921153301833,
173 -0.00002172917966339623434407978263,
174 5.9914810727868915476543145465e-06,
175 -1.5958781571808992162953719817e-06,
176 4.1232986512903717525448312012e-07,
177 -1.0369725993417659101913919101e-07,
178 2.5457982304266541145999235022e-08,
179 -6.1161715053791743082427422443e-09,
180 1.4409346199138658887871461320e-09,
181 -3.3350445956255561668232014995e-10,
182 7.5950686572918996453336138108e-11,
183 -1.7042296334409430377389900278e-11,
184 3.7723525020626230919721640081e-12,
185 -8.2460237635733980528416501227e-13,
186 1.7816961527997797696251868875e-13,
187 -3.8084101506541792942694560802e-14,
188 8.0593669930916099079755351563e-15,
189 -1.6896565961641739017452636964e-15,
190 3.5115651805888443184822853595e-16,
191 -7.2384771938569255638904297651e-17,
192 1.4806598977677176106283840244e-17,
193 -3.0069285750787303634897997963e-18
194 };
195 static cheb_series A3_gt1_cs = {
196 A3_gt1_data,
197 30,
198 -1,1,
199 17
200 };
201
202 /* chebyshev expansion for f(x) = z(x)^8 A_4(z(x)), z(x) = 12/(5(x+1)) */
203 static double A4_gt1_data[30] = {
204 1.15309329391198493586724229008,
205 -1.01812701728669338904729927846,
206 0.71964022270555684403652781941,
207 -0.42359963977172689685150061355,
208 0.215024488759339557817435404261,
209 -0.096751915348145944032096342479,
210 0.039413982058824310099856035361,
211 -0.014775225692561697963781115014,
212 0.005162114514159370516947823271,
213 -0.00169783446445524322560925166335,
214 0.00052995667873006847211519193478,
215 -0.00015802027574996477115667974856,
216 0.000045254366680989687988902825193,
217 -0.000012503722965474638015488600967,
218 3.3457656998119148699124716204e-06,
219 -8.6981575241150758412492331833e-07,
220 2.2030895484325645640823940625e-07,
221 -5.4493369492600677068285936533e-08,
222 1.3190457281724829107139385556e-08,
223 -3.1301560183377379158951191769e-09,
224 7.2937802527123344842593076131e-10,
225 -1.6712080137945140407348940109e-10,
226 3.7700053248213600430503521194e-11,
227 -8.3824538848817227637828899571e-12,
228 1.8388741910049766865274037194e-12,
229 -3.9835919980753778560117573063e-13,
230 8.5288827136546615604290389711e-14,
231 -1.8060227869114416998653266836e-14,
232 3.7849342199690728470461022877e-15,
233 -7.8552867468122209577151823365e-16
234 };
235 static cheb_series A4_gt1_cs = {
236 A4_gt1_data,
237 17, /* 29, */
238 -1, 1,
239 17
240 };
241
242 /* Chebyshev fit for f(x) = z(x)^3 B_2(z(x)), z(x) = 12/(5(x+1)) */
243 static double B2_gt1_data[40] = {
244 0.00118587147272683864479328868589,
245 0.00034820459990648274622193981840,
246 -0.00030411304425639768103075864567,
247 0.00002812066284012343531484682886,
248 0.00004493525295901613184489898748,
249 -0.00003037629997093072196779489677,
250 0.00001125979647123875721949743970,
251 -2.4832533969517775991951008218e-06,
252 -9.9003813640537799587086928278e-08,
253 4.9259859656183110299492296029e-07,
254 -3.7644120964426705960749504975e-07,
255 2.2887828521334625189639122509e-07,
256 -1.3202687370822203731489855050e-07,
257 7.7019669092537400811434860763e-08,
258 -4.6589706973010511603890144294e-08,
259 2.9396476233013923711978522963e-08,
260 -1.9293230611988282919101954538e-08,
261 1.3099107013728717842406906896e-08,
262 -9.1509111940885962831104149355e-09,
263 6.5483472971925614347299375295e-09,
264 -4.7831253582139967461241674569e-09,
265 3.5562625457426178152760148639e-09,
266 -2.6853389444008414186916562103e-09,
267 2.0554738667134200145781857289e-09,
268 -1.5923172019517426277886522758e-09,
269 1.2465923213464381457319481498e-09,
270 -9.8494846881180588507969988989e-10,
271 7.8438674499372126663957464312e-10,
272 -6.2877567918342950225937136855e-10,
273 5.0662318868755257959686944117e-10,
274 -4.0962270881243451160378710952e-10,
275 3.3168684677374908553161911299e-10,
276 -2.6829406619847450633596163305e-10,
277 2.1603988122184568375561077873e-10,
278 -1.7232373309560278402012124481e-10,
279 1.3512709089611470626617830434e-10,
280 -1.0285354732538663013167579792e-10,
281 7.4211345443901713467637018423e-11,
282 -4.8124980266864320351456993068e-11,
283 2.3666534694476306077416831958e-11
284 };
285 static cheb_series B2_gt1_cs = {
286 B2_gt1_data,
287 39,
288 -1, 1,
289 30
290 };
291
292
293 /* Chebyshev fit for f(x) = z(x)^6 B_3(z(x)), z(x) = 12/(5(x+1)) */
294 static double B3_gt1_data[30] = {
295 -0.0102445379362695740863663926486,
296 0.0036618484329295342954730801917,
297 0.0026154252498599303282569321117,
298 -0.0036187389410353156728771706336,
299 0.0021878564157692275944613452462,
300 -0.0008219952303590803584426516821,
301 0.0001281773889155631494321316520,
302 0.0001000944653368032985720548637,
303 -0.0001288293344663774273453147788,
304 0.00010136264202696513867821487205,
305 -0.00007000275849659556221916572733,
306 0.00004694886396757430431607955146,
307 -0.00003190003869717837686356945696,
308 0.00002231453668447775219665947479,
309 -0.00001611102197712439539300336438,
310 0.00001196634424990735214466633513,
311 -9.0986920398931223804111374679e-06,
312 7.0492613694235423068926562567e-06,
313 -5.5425216624642184684300615394e-06,
314 4.4071884714230296614449244106e-06,
315 -3.5328595506791663127928952625e-06,
316 2.84594975572077091520522824686e-06,
317 -2.29592697828824392391071619788e-06,
318 1.84714740375289956396370322228e-06,
319 -1.47383331248116454652025598620e-06,
320 1.15687781098593231076084710267e-06,
321 -8.8174688524627071175315084910e-07,
322 6.3705856964426840441434605593e-07,
323 -4.1358791499961929237755474814e-07,
324 2.0354151158738819867477996807e-07
325 };
326 static cheb_series B3_gt1_cs = {
327 B3_gt1_data,
328 29,
329 -1, 1,
330 29
331 };
332
333
334 /* Chebyshev fit for f(x) = z(x) B_2(z(x)), z(x) = 2(x+1)/5 */
335 static double B2_lt1_data[40] = {
336 0.00073681565841337130021924199490,
337 0.00033803599647571227535304316937,
338 -0.00008251723219239754024210552679,
339 -0.00003390879948656432545900779710,
340 0.00001961398056848881816694014889,
341 -2.35593745904151401624656805567e-06,
342 -1.79055017080406086541563835433e-06,
343 1.33129571185610681090725934031e-06,
344 -5.38879444715436544130673956170e-07,
345 1.49603056041381416881299945557e-07,
346 -1.83377228267274327911131293091e-08,
347 -1.33191430762944336526965187651e-08,
348 1.60642096463700438411396889489e-08,
349 -1.28932576330421806740136816643e-08,
350 9.6169275086179165484403221944e-09,
351 -7.1818502280703532276832887290e-09,
352 5.4744009217215145730697754561e-09,
353 -4.2680446690508456935030086136e-09,
354 3.3941665009266174865683284781e-09,
355 -2.7440714072221673882163135170e-09,
356 2.2488361522108255229193038962e-09,
357 -1.8638240716608748862087923337e-09,
358 1.5592350940805373500866440401e-09,
359 -1.3145743937732330609242633070e-09,
360 1.1153716777215047842790244968e-09,
361 -9.5117576805266622854647303110e-10,
362 8.1428799553234876296804561100e-10,
363 -6.9893770813548773664326279169e-10,
364 6.0073113636087448745018831981e-10,
365 -5.1627434258513453901420776514e-10,
366 4.4290993195074905891788459756e-10,
367 -3.7852978599966867611179315200e-10,
368 3.2143959338863177145307610452e-10,
369 -2.7025926680620777594992221143e-10,
370 2.2384857772457918539228234321e-10,
371 -1.8125071664276678046551271701e-10,
372 1.4164870008713668767293008546e-10,
373 -1.0433101857132782485813325981e-10,
374 6.8663910168392483929411418190e-11,
375 -3.4068313177952244040559740439e-11
376 };
377 static cheb_series B2_lt1_cs = {
378 B2_lt1_data,
379 39,
380 -1, 1,
381 39
382 };
383
384
385 /* Chebyshev fit for f(x) = B_3(2(x+1)/5) */
386 static double B3_lt1_data[40] = {
387 -0.00137160820526992057354001614451,
388 -0.00025474937951101049982680561302,
389 0.00024762975547895881652073467771,
390 0.00005229657281480196749313930265,
391 -0.00007488354272621512385016593760,
392 0.00001416880012891046449980449746,
393 0.00001528986060172183690742576230,
394 -0.00001668672297078590514293325326,
395 0.00001061765189536459018739585094,
396 -5.8220577442406209989680801335e-06,
397 3.3322423743855900506302033234e-06,
398 -2.23292405803003860894449897815e-06,
399 1.74816651036678291794777245325e-06,
400 -1.49581306041395051804547535093e-06,
401 1.32759146107893129050610165582e-06,
402 -1.19376077392564467408373553343e-06,
403 1.07878303863211630544654040875e-06,
404 -9.7743335011819134006676476250e-07,
405 8.8729318903693324226127054792e-07,
406 -8.0671146292125665050876015280e-07,
407 7.3432860378667354971042255937e-07,
408 -6.6897926072697370325310483359e-07,
409 6.0966619703735610352576581485e-07,
410 -5.5554095284507959561958605420e-07,
411 5.0588335673197236002812826526e-07,
412 -4.6008146297767601862670079590e-07,
413 4.1761348515688145911438168306e-07,
414 -3.7803230006989446874174476515e-07,
415 3.4095248501364300041684648230e-07,
416 -3.0603959751354749520615015472e-07,
417 2.7300134179365690589640458993e-07,
418 -2.4158028250762304756044254231e-07,
419 2.1154781038298751985689113868e-07,
420 -1.8269911328756771201465223313e-07,
421 1.5484895085808513749026173074e-07,
422 -1.2782806851555809369226440495e-07,
423 1.0148011725394892565174207341e-07,
424 -7.5658969771439627809239950461e-08,
425 5.0226342286491286957075289622e-08,
426 -2.5049645660259882970547555831e-08
427 };
428 static cheb_series B3_lt1_cs = {
429 B3_lt1_data,
430 39,
431 -1, 1,
432 39
433 };
434
435
436 /* Chebyshev fit for f(x) = A_3(9(x+1)/20) */
437 static double A3_lt1_data[40] = {
438 -0.00017982561472134418587634980117,
439 -0.00036558603837525275836608884064,
440 -0.00002819398055929628850294406363,
441 0.00016704539863875736769812786067,
442 -0.00007098969970347674307623044850,
443 -8.4470843942344237748899879940e-06,
444 0.0000273413090343147765148014327150,
445 -0.0000199073838489821681991178018081,
446 0.0000100004176278235088881096950105,
447 -3.9739852013143676487867902026e-06,
448 1.2265357766449574306882693267e-06,
449 -1.88755584306424047416914864854e-07,
450 -1.37482206060161206336523452036e-07,
451 2.10326379301853336795686477738e-07,
452 -2.05583778245412633433934301948e-07,
453 1.82377384812654863038691147988e-07,
454 -1.58130247846381041027699152436e-07,
455 1.36966982725588978654041029615e-07,
456 -1.19250280944620257443805710485e-07,
457 1.04477169029350256435316644493e-07,
458 -9.2064832489437534542041040184e-08,
459 8.1523798290458784610230199344e-08,
460 -7.2471794980050867512294061891e-08,
461 6.4614432955971132569968860233e-08,
462 -5.7724095125560946811081322985e-08,
463 5.1623107567436835158110947901e-08,
464 -4.6171250746798606260216486042e-08,
465 4.1256621998650164023254101585e-08,
466 -3.6788925543159819135102047082e-08,
467 3.2694499457951844422299750661e-08,
468 -2.89125899697964696586521743928e-08,
469 2.53925288725374047626589488217e-08,
470 -2.20915707933726481321465184207e-08,
471 1.89732166352720474944407102940e-08,
472 -1.60058977893259856012119939554e-08,
473 1.31619294542205876946742394494e-08,
474 -1.04166651771938038563454275883e-08,
475 7.7478015858156185064152078434e-09,
476 -5.1347942579352613057675111787e-09,
477 2.5583541594586723967261504321e-09
478 };
479 static cheb_series A3_lt1_cs = {
480 A3_lt1_data,
481 39,
482 -1, 1,
483 39
484 };
485
486 /* chebyshev fit for f(x) = A_4(2(x+1)/5) */
487 static double A4_lt1_data[30] = {
488 0.00009054703770051610946958226736,
489 0.00033066000498098017589672988293,
490 0.00019737453734363989127226073272,
491 -0.00015490809725932037720034762889,
492 -0.00004514948935538730085479280454,
493 0.00007976881782603940889444573924,
494 -0.00003314566154544740986264993251,
495 -1.88212148790135672249935711657e-06,
496 0.0000114788756505519986352882940648,
497 -9.2263039911196207101468331210e-06,
498 5.1401128250377780476084336340e-06,
499 -2.38418218951722002658891397905e-06,
500 1.00664292214481531598338960828e-06,
501 -4.23224678096490060264249970540e-07,
502 2.00132031535793489976535190025e-07,
503 -1.18689501178886741400633921047e-07,
504 8.7819524319114212999768013738e-08,
505 -7.3964150324206644900787216386e-08,
506 6.5780431507637165113885884236e-08,
507 -5.9651053193022652369837650411e-08,
508 5.4447762662767276209052293773e-08,
509 -4.9802057381568863702541294988e-08,
510 4.5571368194694340198117635845e-08,
511 -4.1682117173547642845382848197e-08,
512 3.8084701352766049815367147717e-08,
513 -3.4740302885185237434662649907e-08,
514 3.1616557064701510611273692060e-08,
515 -2.8685739487689556252374879267e-08,
516 2.5923752117132254429002796600e-08,
517 -2.3309428552190587304662883477e-08
518 };
519 static cheb_series A4_lt1_cs = {
520 A4_lt1_data,
521 29,
522 -1, 1,
523 29
524 };
525
526
olver_B0(double z,double abs_zeta)527 static double olver_B0(double z, double abs_zeta)
528 {
529 if(z < 0.98) {
530 const double t = 1.0/sqrt(1.0-z*z);
531 return -5.0/(48.0*abs_zeta*abs_zeta) + t*(-3.0 + 5.0*t*t)/(24.0*sqrt(abs_zeta));
532 }
533 else if(z < 1.02) {
534 const double a = 1.0-z;
535 const double c0 = 0.0179988721413553309252458658183;
536 const double c1 = 0.0111992982212877614645974276203;
537 const double c2 = 0.0059404069786014304317781160605;
538 const double c3 = 0.0028676724516390040844556450173;
539 const double c4 = 0.0012339189052567271708525111185;
540 const double c5 = 0.0004169250674535178764734660248;
541 const double c6 = 0.0000330173385085949806952777365;
542 const double c7 = -0.0001318076238578203009990106425;
543 const double c8 = -0.0001906870370050847239813945647;
544 return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*(c7 + a*c8)))))));
545 }
546 else {
547 const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z)));
548 return -5.0/(48.0*abs_zeta*abs_zeta) + t*(3.0 + 5.0*t*t)/(24.0*sqrt(abs_zeta));
549 }
550 }
551
552
olver_B1(double z,double abs_zeta)553 static double olver_B1(double z, double abs_zeta)
554 {
555 if(z < 0.88) {
556 const double t = 1.0/sqrt(1.0-z*z);
557 const double t2 = t*t;
558 const double rz = sqrt(abs_zeta);
559 const double z32 = rz*rz*rz;
560 const double z92 = z32*z32*z32;
561 const double term1 = t*t*t * (30375.0 - 369603.0*t2 + 765765.0*t2*t2 - 425425.0*t2*t2*t2)/414720.0;
562 const double term2 = 85085.0/(663552.0*z92);
563 const double term3 = 385.0/110592.*t*(3.0-5.0*t2)/(abs_zeta*abs_zeta*abs_zeta);
564 const double term4 = 5.0/55296.0*t2*(81.0 - 462.0*t2 + 385.0*t2*t2)/z32;
565 return -(term1 + term2 + term3 + term4)/rz;
566 }
567 else if(z < 1.12) {
568 const double a = 1.0-z;
569 const double c0 = -0.00149282953213429172050073403334;
570 const double c1 = -0.00175640941909277865678308358128;
571 const double c2 = -0.00113346148874174912576929663517;
572 const double c3 = -0.00034691090981382974689396961817;
573 const double c4 = 0.00022752516104839243675693256916;
574 const double c5 = 0.00051764145724244846447294636552;
575 const double c6 = 0.00058906174858194233998714243010;
576 const double c7 = 0.00053485514521888073087240392846;
577 const double c8 = 0.00042891792986220150647633418796;
578 const double c9 = 0.00031639765900613633260381972850;
579 const double c10 = 0.00021908147678699592975840749194;
580 return c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*(c9+a*c10)))))))));
581 }
582 else {
583 const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z)));
584 const double t2 = t*t;
585 const double rz = sqrt(abs_zeta);
586 const double z32 = rz*rz*rz;
587 const double z92 = z32*z32*z32;
588 const double term1 = -t2*t * (30375.0 + 369603.0*t2 + 765765.0*t2*t2 + 425425.0*t2*t2*t2)/414720.0;
589 const double term2 = 85085.0/(663552.0*z92);
590 const double term3 = -385.0/110592.0*t*(3.0+5.0*t2)/(abs_zeta*abs_zeta*abs_zeta);
591 const double term4 = 5.0/55296.0*t2*(81.0 + 462.0*t2 + 385.0*t2*t2)/z32;
592 return (term1 + term2 + term3 + term4)/rz;
593 }
594 }
595
596
olver_B2(double z)597 static double olver_B2(double z)
598 {
599 if(z < 0.8) {
600 const double x = 5.0*z/2.0 - 1.0;
601 gsl_sf_result c;
602 cheb_eval_e(&B2_lt1_cs, x, &c);
603 return c.val / z;
604 }
605 else if(z <= 1.2) {
606 const double a = 1.0-z;
607 const double c0 = 0.00055221307672129279005986982501;
608 const double c1 = 0.00089586516310476929281129228969;
609 const double c2 = 0.00067015003441569770883539158863;
610 const double c3 = 0.00010166263361949045682945811828;
611 const double c4 = -0.00044086345133806887291336488582;
612 const double c5 = -0.00073963081508788743392883072523;
613 const double c6 = -0.00076745494377839561259903887331;
614 const double c7 = -0.00060829038106040362291568012663;
615 const double c8 = -0.00037128707528893496121336168683;
616 const double c9 = -0.00014116325105702609866850307176;
617 return c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*c9))))))));
618 }
619 else {
620 const double zi = 1.0/z;
621 const double x = 12.0/5.0 * zi - 1.0;
622 gsl_sf_result c;
623 cheb_eval_e(&B2_gt1_cs, x, &c);
624 return c.val * zi*zi*zi;
625 }
626 }
627
628
olver_B3(double z)629 static double olver_B3(double z)
630 {
631 if(z < 0.8) {
632 const double x = 5.0*z/2.0 - 1.0;
633 gsl_sf_result c;
634 cheb_eval_e(&B3_lt1_cs, x, &c);
635 return c.val;
636 }
637 else if(z < 1.2) {
638 const double a = 1.0-z;
639 const double c0 = -0.00047461779655995980754441833105;
640 const double c1 = -0.00095572913429464297452176811898;
641 const double c2 = -0.00080369634512082892655558133973;
642 const double c3 = -0.00000727921669154784138080600339;
643 const double c4 = 0.00093162500331581345235746518994;
644 const double c5 = 0.00149848796913751497227188612403;
645 const double c6 = 0.00148406039675949727870390426462;
646 return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*c6)))));
647 }
648 else {
649 const double x = 12.0/(5.0*z) - 1.0;
650 const double zi2 = 1.0/(z*z);
651 gsl_sf_result c;
652 cheb_eval_e(&B3_gt1_cs, x, &c);
653 return c.val * zi2*zi2*zi2;
654 }
655 }
656
657
olver_A1(double z,double abs_zeta,double * err)658 static double olver_A1(double z, double abs_zeta, double * err)
659 {
660 if(z < 0.98) {
661 double t = 1.0/sqrt(1.0-z*z);
662 double rz = sqrt(abs_zeta);
663 double t2 = t*t;
664 double term1 = t2*(81.0 - 462.0*t2 + 385.0*t2*t2)/1152.0;
665 double term2 = -455.0/(4608.0*abs_zeta*abs_zeta*abs_zeta);
666 double term3 = 7.0*t*(-3.0 + 5.0*t2)/(1152.0*rz*rz*rz);
667 *err = 2.0 * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2) + fabs(term3));
668 return term1 + term2 + term3;
669 }
670 else if(z < 1.02) {
671 const double a = 1.0-z;
672 const double c0 = -0.00444444444444444444444444444444;
673 const double c1 = -0.00184415584415584415584415584416;
674 const double c2 = 0.00056812076812076812076812076812;
675 const double c3 = 0.00168137865661675185484709294233;
676 const double c4 = 0.00186744042139000122193399504324;
677 const double c5 = 0.00161330105833747826430066790326;
678 const double c6 = 0.00123177312220625816558607537838;
679 const double c7 = 0.00087334711007377573881689318421;
680 const double c8 = 0.00059004942455353250141217015410;
681 const double sum = c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*c8)))))));
682 *err = 2.0 * GSL_DBL_EPSILON * fabs(sum);
683 return sum;
684 }
685 else {
686 const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z)));
687 const double rz = sqrt(abs_zeta);
688 const double t2 = t*t;
689 const double term1 = -t2*(81.0 + 462.0*t2 + 385.0*t2*t2)/1152.0;
690 const double term2 = 455.0/(4608.0*abs_zeta*abs_zeta*abs_zeta);
691 const double term3 = -7.0*t*(3.0 + 5.0*t2)/(1152.0*rz*rz*rz);
692 *err = 2.0 * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2) + fabs(term3));
693 return term1 + term2 + term3;
694 }
695 }
696
697
olver_A2(double z,double abs_zeta)698 static double olver_A2(double z, double abs_zeta)
699 {
700 if(z < 0.88) {
701 double t = 1.0/sqrt(1.0-z*z);
702 double t2 = t*t;
703 double t4 = t2*t2;
704 double t6 = t4*t2;
705 double t8 = t4*t4;
706 double rz = sqrt(abs_zeta);
707 double z3 = abs_zeta*abs_zeta*abs_zeta;
708 double z32 = rz*rz*rz;
709 double z92 = z3*z32;
710 double term1 = t4*(4465125.0 - 94121676.0*t2 + 349922430.0*t4 - 446185740.0*t6 + 185910725.0*t8)/39813120.0;
711 double term2 = -40415375.0/(127401984.0*z3*z3);
712 double term3 = -95095.0/15925248.0*t*(3.0-5.0*t2)/z92;
713 double term4 = -455.0/5308416.0 *t2*(81.0 - 462.0*t2 + 385.0*t4)/z3;
714 double term5 = -7.0/19906560.0*t*t2*(30375.0 - 369603.0*t2 + 765765.0*t4 - 425425.0*t6)/z32;
715 return term1 + term2 + term3 + term4 + term5;
716 }
717 else if(z < 1.12) {
718 double a = 1.0-z;
719 const double c0 = 0.000693735541354588973636592684210;
720 const double c1 = 0.000464483490365843307019777608010;
721 const double c2 = -0.000289036254605598132482570468291;
722 const double c3 = -0.000874764943953712638574497548110;
723 const double c4 = -0.001029716376139865629968584679350;
724 const double c5 = -0.000836857329713810600584714031650;
725 const double c6 = -0.000488910893527218954998270124540;
726 const double c7 = -0.000144236747940817220502256810151;
727 const double c8 = 0.000114363800986163478038576460325;
728 const double c9 = 0.000266806881492777536223944807117;
729 const double c10 = -0.011975517576151069627471048587000;
730 return c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*(c9+a*c10)))))))));
731 }
732 else {
733 const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z)));
734 const double t2 = t*t;
735 const double t4 = t2*t2;
736 const double t6 = t4*t2;
737 const double t8 = t4*t4;
738 const double rz = sqrt(abs_zeta);
739 const double z3 = abs_zeta*abs_zeta*abs_zeta;
740 const double z32 = rz*rz*rz;
741 const double z92 = z3*z32;
742 const double term1 = t4*(4465125.0 + 94121676.0*t2 + 349922430.0*t4 + 446185740.0*t6 + 185910725.0*t8)/39813120.0;
743 const double term2 = -40415375.0/(127401984.0*z3*z3);
744 const double term3 = 95095.0/15925248.0*t*(3.0+5.0*t2)/z92;
745 const double term4 = -455.0/5308416.0 *t2*(81.0 + 462.0*t2 + 385.0*t4)/z3;
746 const double term5 = 7.0/19906560.0*t*t2*(30375.0 + 369603.0*t2 + 765765.0*t4 + 425425.0*t6)/z32;
747 return term1 + term2 + term3 + term4 + term5;
748 }
749 }
750
751
olver_A3(double z)752 static double olver_A3(double z)
753 {
754 if(z < 0.9) {
755 const double x = 20.0*z/9.0 - 1.0;
756 gsl_sf_result c;
757 cheb_eval_e(&A3_lt1_cs, x, &c);
758 return c.val;
759 }
760 else if(z < 1.1) {
761 double a = 1.0-z;
762 const double c0 = -0.000354211971457743840771125759200;
763 const double c1 = -0.000312322527890318832782774881353;
764 const double c2 = 0.000277947465383133980329617631915;
765 const double c3 = 0.000919803044747966977054155192400;
766 const double c4 = 0.001147600388275977640983696906320;
767 const double c5 = 0.000869239326123625742931772044544;
768 const double c6 = 0.000287392257282507334785281718027;
769 return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*c6)))));
770 }
771 else {
772 const double x = 11.0/(5.0*z) - 1.0;
773 const double zi2 = 1.0/(z*z);
774 gsl_sf_result c;
775 cheb_eval_e(&A3_gt1_cs, x, &c);
776 return c.val * zi2*zi2*zi2;
777 }
778 }
779
780
olver_A4(double z)781 static double olver_A4(double z)
782 {
783 if(z < 0.8) {
784 const double x = 5.0*z/2.0 - 1.0;
785 gsl_sf_result c;
786 cheb_eval_e(&A4_lt1_cs, x, &c);
787 return c.val;
788 }
789 else if(z < 1.2) {
790 double a = 1.0-z;
791 const double c0 = 0.00037819419920177291402661228437;
792 const double c1 = 0.00040494390552363233477213857527;
793 const double c2 = -0.00045764735528936113047289344569;
794 const double c3 = -0.00165361044229650225813161341879;
795 const double c4 = -0.00217527517983360049717137015539;
796 const double c5 = -0.00152003287866490735107772795537;
797 return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*c5))));
798 }
799 else {
800 const double x = 12.0/(5.0*z) - 1.0;
801 const double zi2 = 1.0/(z*z);
802 gsl_sf_result c;
803 cheb_eval_e(&A4_gt1_cs, x, &c);
804 return c.val * zi2*zi2*zi2*zi2;
805 }
806 }
807
808 inline
olver_Asum(double nu,double z,double abs_zeta,double * err)809 static double olver_Asum(double nu, double z, double abs_zeta, double * err)
810 {
811 double nu2 = nu*nu;
812 double A1_err;
813 double A1 = olver_A1(z, abs_zeta, &A1_err);
814 double A2 = olver_A2(z, abs_zeta);
815 double A3 = olver_A3(z);
816 double A4 = olver_A4(z);
817 *err = A1_err/nu2 + GSL_DBL_EPSILON;
818 return 1.0 + A1/nu2 + A2/(nu2*nu2) + A3/(nu2*nu2*nu2) + A4/(nu2*nu2*nu2*nu2);
819 }
820
821 inline
olver_Bsum(double nu,double z,double abs_zeta)822 static double olver_Bsum(double nu, double z, double abs_zeta)
823 {
824 double nu2 = nu*nu;
825 double B0 = olver_B0(z, abs_zeta);
826 double B1 = olver_B1(z, abs_zeta);
827 double B2 = olver_B2(z);
828 double B3 = olver_B3(z);
829 return B0 + B1/nu2 + B2/(nu2*nu2) + B3/(nu2*nu2*nu2*nu2);
830 }
831
832
833 /* uniform asymptotic, nu -> Inf, [Abramowitz+Stegun, 9.3.35]
834 *
835 * error:
836 * nu = 2: uniformly good to > 6D
837 * nu = 5: uniformly good to > 8D
838 * nu = 10: uniformly good to > 10D
839 * nu = 20: uniformly good to > 13D
840 *
841 */
gsl_sf_bessel_Jnu_asymp_Olver_e(double nu,double x,gsl_sf_result * result)842 int gsl_sf_bessel_Jnu_asymp_Olver_e(double nu, double x, gsl_sf_result * result)
843 {
844 /* CHECK_POINTER(result) */
845
846 if(x <= 0.0 || nu <= 0.0) {
847 DOMAIN_ERROR(result);
848 }
849 else {
850 double zeta, abs_zeta;
851 double arg;
852 double pre;
853 double asum, bsum, asum_err;
854 gsl_sf_result ai;
855 gsl_sf_result aip;
856 double z = x/nu;
857 double crnu = pow(nu, 1.0/3.0);
858 double nu3 = nu*nu*nu;
859 double nu11 = nu3*nu3*nu3*nu*nu;
860 int stat_a, stat_ap;
861
862 if(fabs(1.0-z) < 0.02) {
863 const double a = 1.0-z;
864 const double c0 = 1.25992104989487316476721060728;
865 const double c1 = 0.37797631496846194943016318218;
866 const double c2 = 0.230385563409348235843147082474;
867 const double c3 = 0.165909603649648694839821892031;
868 const double c4 = 0.12931387086451008907;
869 const double c5 = 0.10568046188858133991;
870 const double c6 = 0.08916997952268186978;
871 const double c7 = 0.07700014900618802456;
872 pre = c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*c7))))));
873 zeta = a * pre;
874 pre = sqrt(2.0*sqrt(pre/(1.0+z)));
875 abs_zeta = fabs(zeta);
876 }
877 else if(z < 1.0) {
878 double rt = sqrt(1.0 - z*z);
879 abs_zeta = pow(1.5*(log((1.0+rt)/z) - rt), 2.0/3.0);
880 zeta = abs_zeta;
881 pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt)));
882 }
883 else {
884 /* z > 1 */
885 double rt = z * sqrt(1.0 - 1.0/(z*z));
886 abs_zeta = pow(1.5*(rt - acos(1.0/z)), 2.0/3.0);
887 zeta = -abs_zeta;
888 pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt)));
889 }
890
891 asum = olver_Asum(nu, z, abs_zeta, &asum_err);
892 bsum = olver_Bsum(nu, z, abs_zeta);
893
894 arg = crnu*crnu * zeta;
895 stat_a = gsl_sf_airy_Ai_e(arg, GSL_MODE_DEFAULT, &ai);
896 stat_ap = gsl_sf_airy_Ai_deriv_e(arg, GSL_MODE_DEFAULT, &aip);
897
898 result->val = pre * (ai.val*asum/crnu + aip.val*bsum/(nu*crnu*crnu));
899 result->err = pre * (ai.err * fabs(asum/crnu));
900 result->err += pre * fabs(ai.val) * asum_err / crnu;
901 result->err += pre * fabs(ai.val * asum) / (crnu*nu11);
902 result->err += 8.0 * GSL_DBL_EPSILON * fabs(result->val);
903
904 return GSL_ERROR_SELECT_2(stat_a, stat_ap);
905 }
906 }
907
908
909 /* uniform asymptotic, nu -> Inf, [Abramowitz+Stegun, 9.3.36]
910 *
911 * error:
912 * nu = 2: uniformly good to > 6D
913 * nu = 5: uniformly good to > 8D
914 * nu = 10: uniformly good to > 10D
915 * nu = 20: uniformly good to > 13D
916 */
gsl_sf_bessel_Ynu_asymp_Olver_e(double nu,double x,gsl_sf_result * result)917 int gsl_sf_bessel_Ynu_asymp_Olver_e(double nu, double x, gsl_sf_result * result)
918 {
919 /* CHECK_POINTER(result) */
920
921 if(x <= 0.0 || nu <= 0.0) {
922 DOMAIN_ERROR(result);
923 }
924 else {
925 double zeta, abs_zeta;
926 double arg;
927 double pre;
928 double asum, bsum, asum_err;
929 gsl_sf_result bi;
930 gsl_sf_result bip;
931 double z = x/nu;
932 double crnu = pow(nu, 1.0/3.0);
933 double nu3 = nu*nu*nu;
934 double nu11 = nu3*nu3*nu3*nu*nu;
935 int stat_b, stat_d;
936
937 if(fabs(1.0-z) < 0.02) {
938 const double a = 1.0-z;
939 const double c0 = 1.25992104989487316476721060728;
940 const double c1 = 0.37797631496846194943016318218;
941 const double c2 = 0.230385563409348235843147082474;
942 const double c3 = 0.165909603649648694839821892031;
943 const double c4 = 0.12931387086451008907;
944 const double c5 = 0.10568046188858133991;
945 const double c6 = 0.08916997952268186978;
946 const double c7 = 0.07700014900618802456;
947 pre = c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*c7))))));
948 zeta = a * pre;
949 pre = sqrt(2.0*sqrt(pre/(1.0+z)));
950 abs_zeta = fabs(zeta);
951 }
952 else if(z < 1.0) {
953 double rt = sqrt(1.0 - z*z);
954 abs_zeta = pow(1.5*(log((1.0+rt)/z) - rt), 2.0/3.0);
955 zeta = abs_zeta;
956 pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt)));
957 }
958 else {
959 /* z > 1 */
960 double rt = z * sqrt(1.0 - 1.0/(z*z));
961 double ac = acos(1.0/z);
962 abs_zeta = pow(1.5*(rt - ac), 2.0/3.0);
963 zeta = -abs_zeta;
964 pre = sqrt(2.0*sqrt(abs_zeta)/rt);
965 }
966
967 asum = olver_Asum(nu, z, abs_zeta, &asum_err);
968 bsum = olver_Bsum(nu, z, abs_zeta);
969
970 arg = crnu*crnu * zeta;
971 stat_b = gsl_sf_airy_Bi_e(arg, GSL_MODE_DEFAULT, &bi);
972 stat_d = gsl_sf_airy_Bi_deriv_e(arg, GSL_MODE_DEFAULT, &bip);
973
974 result->val = -pre * (bi.val*asum/crnu + bip.val*bsum/(nu*crnu*crnu));
975 result->err = pre * (bi.err * fabs(asum/crnu));
976 result->err += pre * fabs(bi.val) * asum_err / crnu;
977 result->err += pre * fabs(bi.val*asum) / (crnu*nu11);
978 result->err += 8.0 * GSL_DBL_EPSILON * fabs(result->val);
979
980 return GSL_ERROR_SELECT_2(stat_b, stat_d);
981 }
982 }
983