diff options
Diffstat (limited to 'gsl-1.9/specfunc/bessel_olver.c')
-rw-r--r-- | gsl-1.9/specfunc/bessel_olver.c | 982 |
1 files changed, 982 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/bessel_olver.c b/gsl-1.9/specfunc/bessel_olver.c new file mode 100644 index 0000000..f7be551 --- /dev/null +++ b/gsl-1.9/specfunc/bessel_olver.c @@ -0,0 +1,982 @@ +/* specfunc/bessel_olver.c + * + * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman + * + * This program is free software; you can redistribute it and/or modify + * it under the terms of the GNU General Public License as published by + * the Free Software Foundation; either version 2 of the License, or (at + * your option) any later version. + * + * This program is distributed in the hope that it will be useful, but + * WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + * General Public License for more details. + * + * You should have received a copy of the GNU General Public License + * along with this program; if not, write to the Free Software + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. + */ + +/* Author: G. Jungman */ + +#include <config.h> +#include <gsl/gsl_math.h> +#include <gsl/gsl_errno.h> +#include <gsl/gsl_sf_airy.h> + +#include "error.h" + +#include "bessel.h" +#include "bessel_olver.h" + +#include "chebyshev.h" +#include "cheb_eval.c" + +/* fit for f(x) = zofmzeta((x+1)/2), 0 <= mzeta <= 1 */ +static double zofmzeta_a_data[20] = { + 2.9332563730829348990, + 0.4896518224847036624, + 0.0228637617355380860, + -0.0001715731377284693, + -0.0000105927538148751, + 1.0595602530419e-6, + -4.68016051691e-8, + 5.8310020e-12, + 1.766537581e-10, + -1.45034640e-11, + 4.357772e-13, + 4.60971e-14, + -2.57571e-14, + 2.26468e-14, + -2.22053e-14, + 2.08593e-14, + -1.84454e-14, + 1.50150e-14, + -1.06506e-14, + 5.5375e-15 +}; +static cheb_series zofmzeta_a_cs = { + zofmzeta_a_data, + 19, + -1,1, + 8 +}; + + +/* fit for f(x) = zofmzeta((9x+11)/2), 1 <= mzeta <= 10 */ +static double zofmzeta_b_data[30] = { + 22.40725276466303489, + 10.39808258825165581, + 1.092050144486018425, + -0.071111274777921604, + 0.008990125336059704, + -0.001201950338088875, + 0.000106686807968315, + 0.000017406491576830, + -0.000014946669657805, + 6.189984487752e-6, + -2.049466715178e-6, + 5.87189458020e-7, + -1.46077514157e-7, + 2.9803936132e-8, + -3.817692108e-9, + -4.66980416e-10, + 5.83860334e-10, + -2.78825299e-10, + 1.01682688e-10, + -3.1209928e-11, + 8.111122e-12, + -1.663986e-12, + 1.81364e-13, + 5.3414e-14, + -4.7234e-14, + 2.1689e-14, + -7.815e-15, + 2.371e-15, + -6.04e-16, + 1.20e-16 +}; +static cheb_series zofmzeta_b_cs = { + zofmzeta_b_data, + 29, + -1,1, + 15 +}; + + +/* fit for f(x) = zofmzeta(mz(x))/mz(x)^(3/2), + * mz(x) = (2/(x+1))^(2/3) 10 + * 10 <= mzeta <= Inf + */ +static double zofmzeta_c_data[11] = { + 1.3824761227122911500, + 0.0244856101686774245, + -0.0000842866496282540, + 1.4656076569771e-6, + -3.14874099476e-8, + 7.561134833e-10, + -1.94531643e-11, + 5.245878e-13, + -1.46380e-14, + 4.192e-16, + -1.23e-17 +}; +static cheb_series zofmzeta_c_cs = { + zofmzeta_c_data, + 10, + -1,1, + 6 +}; + + +/* Invert [Abramowitz+Stegun, 9.3.39]. + * Assumes minus_zeta >= 0. + */ +double +gsl_sf_bessel_Olver_zofmzeta(double minus_zeta) +{ + if(minus_zeta < 1.0) { + const double x = 2.0*minus_zeta - 1.0; + gsl_sf_result c; + cheb_eval_e(&zofmzeta_a_cs, x, &c); + return c.val; + } + else if(minus_zeta < 10.0) { + const double x = (2.0*minus_zeta - 11.0)/9.0; + gsl_sf_result c; + cheb_eval_e(&zofmzeta_b_cs, x, &c); + return c.val; + } + else { + const double TEN_32 = 31.62277660168379332; /* 10^(3/2) */ + const double p = pow(minus_zeta, 3.0/2.0); + const double x = 2.0*TEN_32/p - 1.0; + gsl_sf_result c; + cheb_eval_e(&zofmzeta_c_cs, x, &c); + return c.val * p; + } +} + + +/* Chebyshev fit for f(x) = z(x)^6 A_3(z(x)), z(x) = 22/(10(x+1)) */ +static double A3_gt1_data[31] = { + -0.123783199829515294670493131190, + 0.104636462534700704670877382304, + -0.067500816575851826744877535903, + 0.035563362418888483652711005520, + -0.0160738524035979408472979609051, + 0.0064497878252851092073278056238, + -0.00235408261133449663958121821593, + 0.00079545702851302155411892534965, + -0.00025214920745855079895784825637, + 0.00007574004596069392921153301833, + -0.00002172917966339623434407978263, + 5.9914810727868915476543145465e-06, + -1.5958781571808992162953719817e-06, + 4.1232986512903717525448312012e-07, + -1.0369725993417659101913919101e-07, + 2.5457982304266541145999235022e-08, + -6.1161715053791743082427422443e-09, + 1.4409346199138658887871461320e-09, + -3.3350445956255561668232014995e-10, + 7.5950686572918996453336138108e-11, + -1.7042296334409430377389900278e-11, + 3.7723525020626230919721640081e-12, + -8.2460237635733980528416501227e-13, + 1.7816961527997797696251868875e-13, + -3.8084101506541792942694560802e-14, + 8.0593669930916099079755351563e-15, + -1.6896565961641739017452636964e-15, + 3.5115651805888443184822853595e-16, + -7.2384771938569255638904297651e-17, + 1.4806598977677176106283840244e-17, + -3.0069285750787303634897997963e-18 +}; +static cheb_series A3_gt1_cs = { + A3_gt1_data, + 30, + -1,1, + 17 +}; + +/* chebyshev expansion for f(x) = z(x)^8 A_4(z(x)), z(x) = 12/(5(x+1)) */ +static double A4_gt1_data[30] = { + 1.15309329391198493586724229008, + -1.01812701728669338904729927846, + 0.71964022270555684403652781941, + -0.42359963977172689685150061355, + 0.215024488759339557817435404261, + -0.096751915348145944032096342479, + 0.039413982058824310099856035361, + -0.014775225692561697963781115014, + 0.005162114514159370516947823271, + -0.00169783446445524322560925166335, + 0.00052995667873006847211519193478, + -0.00015802027574996477115667974856, + 0.000045254366680989687988902825193, + -0.000012503722965474638015488600967, + 3.3457656998119148699124716204e-06, + -8.6981575241150758412492331833e-07, + 2.2030895484325645640823940625e-07, + -5.4493369492600677068285936533e-08, + 1.3190457281724829107139385556e-08, + -3.1301560183377379158951191769e-09, + 7.2937802527123344842593076131e-10, + -1.6712080137945140407348940109e-10, + 3.7700053248213600430503521194e-11, + -8.3824538848817227637828899571e-12, + 1.8388741910049766865274037194e-12, + -3.9835919980753778560117573063e-13, + 8.5288827136546615604290389711e-14, + -1.8060227869114416998653266836e-14, + 3.7849342199690728470461022877e-15, + -7.8552867468122209577151823365e-16 +}; +static cheb_series A4_gt1_cs = { + A4_gt1_data, + 17, /* 29, */ + -1, 1, + 17 +}; + +/* Chebyshev fit for f(x) = z(x)^3 B_2(z(x)), z(x) = 12/(5(x+1)) */ +static double B2_gt1_data[40] = { + 0.00118587147272683864479328868589, + 0.00034820459990648274622193981840, + -0.00030411304425639768103075864567, + 0.00002812066284012343531484682886, + 0.00004493525295901613184489898748, + -0.00003037629997093072196779489677, + 0.00001125979647123875721949743970, + -2.4832533969517775991951008218e-06, + -9.9003813640537799587086928278e-08, + 4.9259859656183110299492296029e-07, + -3.7644120964426705960749504975e-07, + 2.2887828521334625189639122509e-07, + -1.3202687370822203731489855050e-07, + 7.7019669092537400811434860763e-08, + -4.6589706973010511603890144294e-08, + 2.9396476233013923711978522963e-08, + -1.9293230611988282919101954538e-08, + 1.3099107013728717842406906896e-08, + -9.1509111940885962831104149355e-09, + 6.5483472971925614347299375295e-09, + -4.7831253582139967461241674569e-09, + 3.5562625457426178152760148639e-09, + -2.6853389444008414186916562103e-09, + 2.0554738667134200145781857289e-09, + -1.5923172019517426277886522758e-09, + 1.2465923213464381457319481498e-09, + -9.8494846881180588507969988989e-10, + 7.8438674499372126663957464312e-10, + -6.2877567918342950225937136855e-10, + 5.0662318868755257959686944117e-10, + -4.0962270881243451160378710952e-10, + 3.3168684677374908553161911299e-10, + -2.6829406619847450633596163305e-10, + 2.1603988122184568375561077873e-10, + -1.7232373309560278402012124481e-10, + 1.3512709089611470626617830434e-10, + -1.0285354732538663013167579792e-10, + 7.4211345443901713467637018423e-11, + -4.8124980266864320351456993068e-11, + 2.3666534694476306077416831958e-11 +}; +static cheb_series B2_gt1_cs = { + B2_gt1_data, + 39, + -1, 1, + 30 +}; + + +/* Chebyshev fit for f(x) = z(x)^6 B_3(z(x)), z(x) = 12/(5(x+1)) */ +static double B3_gt1_data[30] = { + -0.0102445379362695740863663926486, + 0.0036618484329295342954730801917, + 0.0026154252498599303282569321117, + -0.0036187389410353156728771706336, + 0.0021878564157692275944613452462, + -0.0008219952303590803584426516821, + 0.0001281773889155631494321316520, + 0.0001000944653368032985720548637, + -0.0001288293344663774273453147788, + 0.00010136264202696513867821487205, + -0.00007000275849659556221916572733, + 0.00004694886396757430431607955146, + -0.00003190003869717837686356945696, + 0.00002231453668447775219665947479, + -0.00001611102197712439539300336438, + 0.00001196634424990735214466633513, + -9.0986920398931223804111374679e-06, + 7.0492613694235423068926562567e-06, + -5.5425216624642184684300615394e-06, + 4.4071884714230296614449244106e-06, + -3.5328595506791663127928952625e-06, + 2.84594975572077091520522824686e-06, + -2.29592697828824392391071619788e-06, + 1.84714740375289956396370322228e-06, + -1.47383331248116454652025598620e-06, + 1.15687781098593231076084710267e-06, + -8.8174688524627071175315084910e-07, + 6.3705856964426840441434605593e-07, + -4.1358791499961929237755474814e-07, + 2.0354151158738819867477996807e-07 +}; +static cheb_series B3_gt1_cs = { + B3_gt1_data, + 29, + -1, 1, + 29 +}; + + +/* Chebyshev fit for f(x) = z(x) B_2(z(x)), z(x) = 2(x+1)/5 */ +static double B2_lt1_data[40] = { + 0.00073681565841337130021924199490, + 0.00033803599647571227535304316937, + -0.00008251723219239754024210552679, + -0.00003390879948656432545900779710, + 0.00001961398056848881816694014889, + -2.35593745904151401624656805567e-06, + -1.79055017080406086541563835433e-06, + 1.33129571185610681090725934031e-06, + -5.38879444715436544130673956170e-07, + 1.49603056041381416881299945557e-07, + -1.83377228267274327911131293091e-08, + -1.33191430762944336526965187651e-08, + 1.60642096463700438411396889489e-08, + -1.28932576330421806740136816643e-08, + 9.6169275086179165484403221944e-09, + -7.1818502280703532276832887290e-09, + 5.4744009217215145730697754561e-09, + -4.2680446690508456935030086136e-09, + 3.3941665009266174865683284781e-09, + -2.7440714072221673882163135170e-09, + 2.2488361522108255229193038962e-09, + -1.8638240716608748862087923337e-09, + 1.5592350940805373500866440401e-09, + -1.3145743937732330609242633070e-09, + 1.1153716777215047842790244968e-09, + -9.5117576805266622854647303110e-10, + 8.1428799553234876296804561100e-10, + -6.9893770813548773664326279169e-10, + 6.0073113636087448745018831981e-10, + -5.1627434258513453901420776514e-10, + 4.4290993195074905891788459756e-10, + -3.7852978599966867611179315200e-10, + 3.2143959338863177145307610452e-10, + -2.7025926680620777594992221143e-10, + 2.2384857772457918539228234321e-10, + -1.8125071664276678046551271701e-10, + 1.4164870008713668767293008546e-10, + -1.0433101857132782485813325981e-10, + 6.8663910168392483929411418190e-11, + -3.4068313177952244040559740439e-11 +}; +static cheb_series B2_lt1_cs = { + B2_lt1_data, + 39, + -1, 1, + 39 +}; + + +/* Chebyshev fit for f(x) = B_3(2(x+1)/5) */ +static double B3_lt1_data[40] = { + -0.00137160820526992057354001614451, + -0.00025474937951101049982680561302, + 0.00024762975547895881652073467771, + 0.00005229657281480196749313930265, + -0.00007488354272621512385016593760, + 0.00001416880012891046449980449746, + 0.00001528986060172183690742576230, + -0.00001668672297078590514293325326, + 0.00001061765189536459018739585094, + -5.8220577442406209989680801335e-06, + 3.3322423743855900506302033234e-06, + -2.23292405803003860894449897815e-06, + 1.74816651036678291794777245325e-06, + -1.49581306041395051804547535093e-06, + 1.32759146107893129050610165582e-06, + -1.19376077392564467408373553343e-06, + 1.07878303863211630544654040875e-06, + -9.7743335011819134006676476250e-07, + 8.8729318903693324226127054792e-07, + -8.0671146292125665050876015280e-07, + 7.3432860378667354971042255937e-07, + -6.6897926072697370325310483359e-07, + 6.0966619703735610352576581485e-07, + -5.5554095284507959561958605420e-07, + 5.0588335673197236002812826526e-07, + -4.6008146297767601862670079590e-07, + 4.1761348515688145911438168306e-07, + -3.7803230006989446874174476515e-07, + 3.4095248501364300041684648230e-07, + -3.0603959751354749520615015472e-07, + 2.7300134179365690589640458993e-07, + -2.4158028250762304756044254231e-07, + 2.1154781038298751985689113868e-07, + -1.8269911328756771201465223313e-07, + 1.5484895085808513749026173074e-07, + -1.2782806851555809369226440495e-07, + 1.0148011725394892565174207341e-07, + -7.5658969771439627809239950461e-08, + 5.0226342286491286957075289622e-08, + -2.5049645660259882970547555831e-08 +}; +static cheb_series B3_lt1_cs = { + B3_lt1_data, + 39, + -1, 1, + 39 +}; + + +/* Chebyshev fit for f(x) = A_3(9(x+1)/20) */ +static double A3_lt1_data[40] = { + -0.00017982561472134418587634980117, + -0.00036558603837525275836608884064, + -0.00002819398055929628850294406363, + 0.00016704539863875736769812786067, + -0.00007098969970347674307623044850, + -8.4470843942344237748899879940e-06, + 0.0000273413090343147765148014327150, + -0.0000199073838489821681991178018081, + 0.0000100004176278235088881096950105, + -3.9739852013143676487867902026e-06, + 1.2265357766449574306882693267e-06, + -1.88755584306424047416914864854e-07, + -1.37482206060161206336523452036e-07, + 2.10326379301853336795686477738e-07, + -2.05583778245412633433934301948e-07, + 1.82377384812654863038691147988e-07, + -1.58130247846381041027699152436e-07, + 1.36966982725588978654041029615e-07, + -1.19250280944620257443805710485e-07, + 1.04477169029350256435316644493e-07, + -9.2064832489437534542041040184e-08, + 8.1523798290458784610230199344e-08, + -7.2471794980050867512294061891e-08, + 6.4614432955971132569968860233e-08, + -5.7724095125560946811081322985e-08, + 5.1623107567436835158110947901e-08, + -4.6171250746798606260216486042e-08, + 4.1256621998650164023254101585e-08, + -3.6788925543159819135102047082e-08, + 3.2694499457951844422299750661e-08, + -2.89125899697964696586521743928e-08, + 2.53925288725374047626589488217e-08, + -2.20915707933726481321465184207e-08, + 1.89732166352720474944407102940e-08, + -1.60058977893259856012119939554e-08, + 1.31619294542205876946742394494e-08, + -1.04166651771938038563454275883e-08, + 7.7478015858156185064152078434e-09, + -5.1347942579352613057675111787e-09, + 2.5583541594586723967261504321e-09 +}; +static cheb_series A3_lt1_cs = { + A3_lt1_data, + 39, + -1, 1, + 39 +}; + +/* chebyshev fit for f(x) = A_4(2(x+1)/5) */ +static double A4_lt1_data[30] = { + 0.00009054703770051610946958226736, + 0.00033066000498098017589672988293, + 0.00019737453734363989127226073272, + -0.00015490809725932037720034762889, + -0.00004514948935538730085479280454, + 0.00007976881782603940889444573924, + -0.00003314566154544740986264993251, + -1.88212148790135672249935711657e-06, + 0.0000114788756505519986352882940648, + -9.2263039911196207101468331210e-06, + 5.1401128250377780476084336340e-06, + -2.38418218951722002658891397905e-06, + 1.00664292214481531598338960828e-06, + -4.23224678096490060264249970540e-07, + 2.00132031535793489976535190025e-07, + -1.18689501178886741400633921047e-07, + 8.7819524319114212999768013738e-08, + -7.3964150324206644900787216386e-08, + 6.5780431507637165113885884236e-08, + -5.9651053193022652369837650411e-08, + 5.4447762662767276209052293773e-08, + -4.9802057381568863702541294988e-08, + 4.5571368194694340198117635845e-08, + -4.1682117173547642845382848197e-08, + 3.8084701352766049815367147717e-08, + -3.4740302885185237434662649907e-08, + 3.1616557064701510611273692060e-08, + -2.8685739487689556252374879267e-08, + 2.5923752117132254429002796600e-08, + -2.3309428552190587304662883477e-08 +}; +static cheb_series A4_lt1_cs = { + A4_lt1_data, + 29, + -1, 1, + 29 +}; + + +static double olver_B0(double z, double abs_zeta) +{ + if(z < 0.98) { + const double t = 1.0/sqrt(1.0-z*z); + return -5.0/(48.0*abs_zeta*abs_zeta) + t*(-3.0 + 5.0*t*t)/(24.0*sqrt(abs_zeta)); + } + else if(z < 1.02) { + const double a = 1.0-z; + const double c0 = 0.0179988721413553309252458658183; + const double c1 = 0.0111992982212877614645974276203; + const double c2 = 0.0059404069786014304317781160605; + const double c3 = 0.0028676724516390040844556450173; + const double c4 = 0.0012339189052567271708525111185; + const double c5 = 0.0004169250674535178764734660248; + const double c6 = 0.0000330173385085949806952777365; + const double c7 = -0.0001318076238578203009990106425; + const double c8 = -0.0001906870370050847239813945647; + return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*(c7 + a*c8))))))); + } + else { + const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z))); + return -5.0/(48.0*abs_zeta*abs_zeta) + t*(3.0 + 5.0*t*t)/(24.0*sqrt(abs_zeta)); + } +} + + +static double olver_B1(double z, double abs_zeta) +{ + if(z < 0.88) { + const double t = 1.0/sqrt(1.0-z*z); + const double t2 = t*t; + const double rz = sqrt(abs_zeta); + const double z32 = rz*rz*rz; + const double z92 = z32*z32*z32; + const double term1 = t*t*t * (30375.0 - 369603.0*t2 + 765765.0*t2*t2 - 425425.0*t2*t2*t2)/414720.0; + const double term2 = 85085.0/(663552.0*z92); + const double term3 = 385.0/110592.*t*(3.0-5.0*t2)/(abs_zeta*abs_zeta*abs_zeta); + const double term4 = 5.0/55296.0*t2*(81.0 - 462.0*t2 + 385.0*t2*t2)/z32; + return -(term1 + term2 + term3 + term4)/rz; + } + else if(z < 1.12) { + const double a = 1.0-z; + const double c0 = -0.00149282953213429172050073403334; + const double c1 = -0.00175640941909277865678308358128; + const double c2 = -0.00113346148874174912576929663517; + const double c3 = -0.00034691090981382974689396961817; + const double c4 = 0.00022752516104839243675693256916; + const double c5 = 0.00051764145724244846447294636552; + const double c6 = 0.00058906174858194233998714243010; + const double c7 = 0.00053485514521888073087240392846; + const double c8 = 0.00042891792986220150647633418796; + const double c9 = 0.00031639765900613633260381972850; + const double c10 = 0.00021908147678699592975840749194; + return c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*(c9+a*c10))))))))); + } + else { + const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z))); + const double t2 = t*t; + const double rz = sqrt(abs_zeta); + const double z32 = rz*rz*rz; + const double z92 = z32*z32*z32; + const double term1 = -t2*t * (30375.0 + 369603.0*t2 + 765765.0*t2*t2 + 425425.0*t2*t2*t2)/414720.0; + const double term2 = 85085.0/(663552.0*z92); + const double term3 = -385.0/110592.0*t*(3.0+5.0*t2)/(abs_zeta*abs_zeta*abs_zeta); + const double term4 = 5.0/55296.0*t2*(81.0 + 462.0*t2 + 385.0*t2*t2)/z32; + return (term1 + term2 + term3 + term4)/rz; + } +} + + +static double olver_B2(double z, double abs_zeta) +{ + if(z < 0.8) { + const double x = 5.0*z/2.0 - 1.0; + gsl_sf_result c; + cheb_eval_e(&B2_lt1_cs, x, &c); + return c.val / z; + } + else if(z <= 1.2) { + const double a = 1.0-z; + const double c0 = 0.00055221307672129279005986982501; + const double c1 = 0.00089586516310476929281129228969; + const double c2 = 0.00067015003441569770883539158863; + const double c3 = 0.00010166263361949045682945811828; + const double c4 = -0.00044086345133806887291336488582; + const double c5 = -0.00073963081508788743392883072523; + const double c6 = -0.00076745494377839561259903887331; + const double c7 = -0.00060829038106040362291568012663; + const double c8 = -0.00037128707528893496121336168683; + const double c9 = -0.00014116325105702609866850307176; + return c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*c9)))))))); + } + else { + const double zi = 1.0/z; + const double x = 12.0/5.0 * zi - 1.0; + gsl_sf_result c; + cheb_eval_e(&B2_gt1_cs, x, &c); + return c.val * zi*zi*zi; + } +} + + +static double olver_B3(double z, double abs_zeta) +{ + if(z < 0.8) { + const double x = 5.0*z/2.0 - 1.0; + gsl_sf_result c; + cheb_eval_e(&B3_lt1_cs, x, &c); + return c.val; + } + else if(z < 1.2) { + const double a = 1.0-z; + const double c0 = -0.00047461779655995980754441833105; + const double c1 = -0.00095572913429464297452176811898; + const double c2 = -0.00080369634512082892655558133973; + const double c3 = -0.00000727921669154784138080600339; + const double c4 = 0.00093162500331581345235746518994; + const double c5 = 0.00149848796913751497227188612403; + const double c6 = 0.00148406039675949727870390426462; + return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*c6))))); + } + else { + const double x = 12.0/(5.0*z) - 1.0; + const double zi2 = 1.0/(z*z); + gsl_sf_result c; + cheb_eval_e(&B3_gt1_cs, x, &c); + return c.val * zi2*zi2*zi2; + } +} + + +static double olver_A1(double z, double abs_zeta, double * err) +{ + if(z < 0.98) { + double t = 1.0/sqrt(1.0-z*z); + double rz = sqrt(abs_zeta); + double t2 = t*t; + double term1 = t2*(81.0 - 462.0*t2 + 385.0*t2*t2)/1152.0; + double term2 = -455.0/(4608.0*abs_zeta*abs_zeta*abs_zeta); + double term3 = 7.0*t*(-3.0 + 5.0*t2)/(1152.0*rz*rz*rz); + *err = 2.0 * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2) + fabs(term3)); + return term1 + term2 + term3; + } + else if(z < 1.02) { + const double a = 1.0-z; + const double c0 = -0.00444444444444444444444444444444; + const double c1 = -0.00184415584415584415584415584416; + const double c2 = 0.00056812076812076812076812076812; + const double c3 = 0.00168137865661675185484709294233; + const double c4 = 0.00186744042139000122193399504324; + const double c5 = 0.00161330105833747826430066790326; + const double c6 = 0.00123177312220625816558607537838; + const double c7 = 0.00087334711007377573881689318421; + const double c8 = 0.00059004942455353250141217015410; + const double sum = c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*c8))))))); + *err = 2.0 * GSL_DBL_EPSILON * fabs(sum); + return sum; + } + else { + const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z))); + const double rz = sqrt(abs_zeta); + const double t2 = t*t; + const double term1 = -t2*(81.0 + 462.0*t2 + 385.0*t2*t2)/1152.0; + const double term2 = 455.0/(4608.0*abs_zeta*abs_zeta*abs_zeta); + const double term3 = -7.0*t*(3.0 + 5.0*t2)/(1152.0*rz*rz*rz); + *err = 2.0 * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2) + fabs(term3)); + return term1 + term2 + term3; + } +} + + +static double olver_A2(double z, double abs_zeta) +{ + if(z < 0.88) { + double t = 1.0/sqrt(1.0-z*z); + double t2 = t*t; + double t4 = t2*t2; + double t6 = t4*t2; + double t8 = t4*t4; + double rz = sqrt(abs_zeta); + double z3 = abs_zeta*abs_zeta*abs_zeta; + double z32 = rz*rz*rz; + double z92 = z3*z32; + double term1 = t4*(4465125.0 - 94121676.0*t2 + 349922430.0*t4 - 446185740.0*t6 + 185910725.0*t8)/39813120.0; + double term2 = -40415375.0/(127401984.0*z3*z3); + double term3 = -95095.0/15925248.0*t*(3.0-5.0*t2)/z92; + double term4 = -455.0/5308416.0 *t2*(81.0 - 462.0*t2 + 385.0*t4)/z3; + double term5 = -7.0/19906560.0*t*t2*(30375.0 - 369603.0*t2 + 765765.0*t4 - 425425.0*t6)/z32; + return term1 + term2 + term3 + term4 + term5; + } + else if(z < 1.12) { + double a = 1.0-z; + const double c0 = 0.000693735541354588973636592684210; + const double c1 = 0.000464483490365843307019777608010; + const double c2 = -0.000289036254605598132482570468291; + const double c3 = -0.000874764943953712638574497548110; + const double c4 = -0.001029716376139865629968584679350; + const double c5 = -0.000836857329713810600584714031650; + const double c6 = -0.000488910893527218954998270124540; + const double c7 = -0.000144236747940817220502256810151; + const double c8 = 0.000114363800986163478038576460325; + const double c9 = 0.000266806881492777536223944807117; + const double c10 = -0.011975517576151069627471048587000; + return c0+a*(c1+a*(c2+a*(c3+a*(c4+a*(c5+a*(c6+a*(c7+a*(c8+a*(c9+a*c10))))))))); + } + else { + const double t = 1.0/(z*sqrt(1.0 - 1.0/(z*z))); + const double t2 = t*t; + const double t4 = t2*t2; + const double t6 = t4*t2; + const double t8 = t4*t4; + const double rz = sqrt(abs_zeta); + const double z3 = abs_zeta*abs_zeta*abs_zeta; + const double z32 = rz*rz*rz; + const double z92 = z3*z32; + const double term1 = t4*(4465125.0 + 94121676.0*t2 + 349922430.0*t4 + 446185740.0*t6 + 185910725.0*t8)/39813120.0; + const double term2 = -40415375.0/(127401984.0*z3*z3); + const double term3 = 95095.0/15925248.0*t*(3.0+5.0*t2)/z92; + const double term4 = -455.0/5308416.0 *t2*(81.0 + 462.0*t2 + 385.0*t4)/z3; + const double term5 = 7.0/19906560.0*t*t2*(30375.0 + 369603.0*t2 + 765765.0*t4 + 425425.0*t6)/z32; + return term1 + term2 + term3 + term4 + term5; + } +} + + +static double olver_A3(double z, double abs_zeta) +{ + if(z < 0.9) { + const double x = 20.0*z/9.0 - 1.0; + gsl_sf_result c; + cheb_eval_e(&A3_lt1_cs, x, &c); + return c.val; + } + else if(z < 1.1) { + double a = 1.0-z; + const double c0 = -0.000354211971457743840771125759200; + const double c1 = -0.000312322527890318832782774881353; + const double c2 = 0.000277947465383133980329617631915; + const double c3 = 0.000919803044747966977054155192400; + const double c4 = 0.001147600388275977640983696906320; + const double c5 = 0.000869239326123625742931772044544; + const double c6 = 0.000287392257282507334785281718027; + return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*c6))))); + } + else { + const double x = 11.0/(5.0*z) - 1.0; + const double zi2 = 1.0/(z*z); + gsl_sf_result c; + cheb_eval_e(&A3_gt1_cs, x, &c); + return c.val * zi2*zi2*zi2; + } +} + + +static double olver_A4(double z, double abs_zeta) +{ + if(z < 0.8) { + const double x = 5.0*z/2.0 - 1.0; + gsl_sf_result c; + cheb_eval_e(&A4_lt1_cs, x, &c); + return c.val; + } + else if(z < 1.2) { + double a = 1.0-z; + const double c0 = 0.00037819419920177291402661228437; + const double c1 = 0.00040494390552363233477213857527; + const double c2 = -0.00045764735528936113047289344569; + const double c3 = -0.00165361044229650225813161341879; + const double c4 = -0.00217527517983360049717137015539; + const double c5 = -0.00152003287866490735107772795537; + return c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*c5)))); + } + else { + const double x = 12.0/(5.0*z) - 1.0; + const double zi2 = 1.0/(z*z); + gsl_sf_result c; + cheb_eval_e(&A4_gt1_cs, x, &c); + return c.val * zi2*zi2*zi2*zi2; + } +} + +inline +static double olver_Asum(double nu, double z, double abs_zeta, double * err) +{ + double nu2 = nu*nu; + double A1_err; + double A1 = olver_A1(z, abs_zeta, &A1_err); + double A2 = olver_A2(z, abs_zeta); + double A3 = olver_A3(z, abs_zeta); + double A4 = olver_A4(z, abs_zeta); + *err = A1_err/nu2 + GSL_DBL_EPSILON; + return 1.0 + A1/nu2 + A2/(nu2*nu2) + A3/(nu2*nu2*nu2) + A4/(nu2*nu2*nu2*nu2); +} + +inline +static double olver_Bsum(double nu, double z, double abs_zeta) +{ + double nu2 = nu*nu; + double B0 = olver_B0(z, abs_zeta); + double B1 = olver_B1(z, abs_zeta); + double B2 = olver_B2(z, abs_zeta); + double B3 = olver_B3(z, abs_zeta); + return B0 + B1/nu2 + B2/(nu2*nu2) + B3/(nu2*nu2*nu2*nu2); +} + + +/* uniform asymptotic, nu -> Inf, [Abramowitz+Stegun, 9.3.35] + * + * error: + * nu = 2: uniformly good to > 6D + * nu = 5: uniformly good to > 8D + * nu = 10: uniformly good to > 10D + * nu = 20: uniformly good to > 13D + * + */ +int gsl_sf_bessel_Jnu_asymp_Olver_e(double nu, double x, gsl_sf_result * result) +{ + /* CHECK_POINTER(result) */ + + if(x <= 0.0 || nu <= 0.0) { + DOMAIN_ERROR(result); + } + else { + double zeta, abs_zeta; + double arg; + double pre; + double asum, bsum, asum_err; + gsl_sf_result ai; + gsl_sf_result aip; + double z = x/nu; + double crnu = pow(nu, 1.0/3.0); + double nu3 = nu*nu*nu; + double nu11 = nu3*nu3*nu3*nu*nu; + int stat_a, stat_ap; + + if(fabs(1.0-z) < 0.02) { + const double a = 1.0-z; + const double c0 = 1.25992104989487316476721060728; + const double c1 = 0.37797631496846194943016318218; + const double c2 = 0.230385563409348235843147082474; + const double c3 = 0.165909603649648694839821892031; + const double c4 = 0.12931387086451008907; + const double c5 = 0.10568046188858133991; + const double c6 = 0.08916997952268186978; + const double c7 = 0.07700014900618802456; + pre = c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*c7)))))); + zeta = a * pre; + pre = sqrt(2.0*sqrt(pre/(1.0+z))); + abs_zeta = fabs(zeta); + } + else if(z < 1.0) { + double rt = sqrt(1.0 - z*z); + abs_zeta = pow(1.5*(log((1.0+rt)/z) - rt), 2.0/3.0); + zeta = abs_zeta; + pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt))); + } + else { + /* z > 1 */ + double rt = z * sqrt(1.0 - 1.0/(z*z)); + abs_zeta = pow(1.5*(rt - acos(1.0/z)), 2.0/3.0); + zeta = -abs_zeta; + pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt))); + } + + asum = olver_Asum(nu, z, abs_zeta, &asum_err); + bsum = olver_Bsum(nu, z, abs_zeta); + + arg = crnu*crnu * zeta; + stat_a = gsl_sf_airy_Ai_e(arg, GSL_MODE_DEFAULT, &ai); + stat_ap = gsl_sf_airy_Ai_deriv_e(arg, GSL_MODE_DEFAULT, &aip); + + result->val = pre * (ai.val*asum/crnu + aip.val*bsum/(nu*crnu*crnu)); + result->err = pre * (ai.err * fabs(asum/crnu)); + result->err += pre * fabs(ai.val) * asum_err / crnu; + result->err += pre * fabs(ai.val * asum) / (crnu*nu11); + result->err += 8.0 * GSL_DBL_EPSILON * fabs(result->val); + + return GSL_ERROR_SELECT_2(stat_a, stat_ap); + } +} + + +/* uniform asymptotic, nu -> Inf, [Abramowitz+Stegun, 9.3.36] + * + * error: + * nu = 2: uniformly good to > 6D + * nu = 5: uniformly good to > 8D + * nu = 10: uniformly good to > 10D + * nu = 20: uniformly good to > 13D + */ +int gsl_sf_bessel_Ynu_asymp_Olver_e(double nu, double x, gsl_sf_result * result) +{ + /* CHECK_POINTER(result) */ + + if(x <= 0.0 || nu <= 0.0) { + DOMAIN_ERROR(result); + } + else { + double zeta, abs_zeta; + double arg; + double pre; + double asum, bsum, asum_err; + gsl_sf_result bi; + gsl_sf_result bip; + double z = x/nu; + double crnu = pow(nu, 1.0/3.0); + double nu3 = nu*nu*nu; + double nu11 = nu3*nu3*nu3*nu*nu; + int stat_b, stat_d; + + if(fabs(1.0-z) < 0.02) { + const double a = 1.0-z; + const double c0 = 1.25992104989487316476721060728; + const double c1 = 0.37797631496846194943016318218; + const double c2 = 0.230385563409348235843147082474; + const double c3 = 0.165909603649648694839821892031; + const double c4 = 0.12931387086451008907; + const double c5 = 0.10568046188858133991; + const double c6 = 0.08916997952268186978; + const double c7 = 0.07700014900618802456; + pre = c0 + a*(c1 + a*(c2 + a*(c3 + a*(c4 + a*(c5 + a*(c6 + a*c7)))))); + zeta = a * pre; + pre = sqrt(2.0*sqrt(pre/(1.0+z))); + abs_zeta = fabs(zeta); + } + else if(z < 1.0) { + double rt = sqrt(1.0 - z*z); + abs_zeta = pow(1.5*(log((1.0+rt)/z) - rt), 2.0/3.0); + zeta = abs_zeta; + pre = sqrt(2.0*sqrt(abs_zeta/(rt*rt))); + } + else { + /* z > 1 */ + double rt = z * sqrt(1.0 - 1.0/(z*z)); + double ac = acos(1.0/z); + abs_zeta = pow(1.5*(rt - ac), 2.0/3.0); + zeta = -abs_zeta; + pre = sqrt(2.0*sqrt(abs_zeta)/rt); + } + + asum = olver_Asum(nu, z, abs_zeta, &asum_err); + bsum = olver_Bsum(nu, z, abs_zeta); + + arg = crnu*crnu * zeta; + stat_b = gsl_sf_airy_Bi_e(arg, GSL_MODE_DEFAULT, &bi); + stat_d = gsl_sf_airy_Bi_deriv_e(arg, GSL_MODE_DEFAULT, &bip); + + result->val = -pre * (bi.val*asum/crnu + bip.val*bsum/(nu*crnu*crnu)); + result->err = pre * (bi.err * fabs(asum/crnu)); + result->err += pre * fabs(bi.val) * asum_err / crnu; + result->err += pre * fabs(bi.val*asum) / (crnu*nu11); + result->err += 8.0 * GSL_DBL_EPSILON * fabs(result->val); + + return GSL_ERROR_SELECT_2(stat_b, stat_d); + } +} |