| /* specfunc/zeta.c |
| * |
| * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004 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_elementary.h> |
| #include <gsl/gsl_sf_exp.h> |
| #include <gsl/gsl_sf_gamma.h> |
| #include <gsl/gsl_sf_pow_int.h> |
| #include <gsl/gsl_sf_zeta.h> |
| |
| #include "error.h" |
| |
| #include "chebyshev.h" |
| #include "cheb_eval.c" |
| |
| #define LogTwoPi_ 1.8378770664093454835606594728111235279723 |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| /* chebyshev fit for (s(t)-1)Zeta[s(t)] |
| * s(t)= (t+1)/2 |
| * -1 <= t <= 1 |
| */ |
| static double zeta_xlt1_data[14] = { |
| 1.48018677156931561235192914649, |
| 0.25012062539889426471999938167, |
| 0.00991137502135360774243761467, |
| -0.00012084759656676410329833091, |
| -4.7585866367662556504652535281e-06, |
| 2.2229946694466391855561441361e-07, |
| -2.2237496498030257121309056582e-09, |
| -1.0173226513229028319420799028e-10, |
| 4.3756643450424558284466248449e-12, |
| -6.2229632593100551465504090814e-14, |
| -6.6116201003272207115277520305e-16, |
| 4.9477279533373912324518463830e-17, |
| -1.0429819093456189719660003522e-18, |
| 6.9925216166580021051464412040e-21, |
| }; |
| static cheb_series zeta_xlt1_cs = { |
| zeta_xlt1_data, |
| 13, |
| -1, 1, |
| 8 |
| }; |
| |
| /* chebyshev fit for (s(t)-1)Zeta[s(t)] |
| * s(t)= (19t+21)/2 |
| * -1 <= t <= 1 |
| */ |
| static double zeta_xgt1_data[30] = { |
| 19.3918515726724119415911269006, |
| 9.1525329692510756181581271500, |
| 0.2427897658867379985365270155, |
| -0.1339000688262027338316641329, |
| 0.0577827064065028595578410202, |
| -0.0187625983754002298566409700, |
| 0.0039403014258320354840823803, |
| -0.0000581508273158127963598882, |
| -0.0003756148907214820704594549, |
| 0.0001892530548109214349092999, |
| -0.0000549032199695513496115090, |
| 8.7086484008939038610413331863e-6, |
| 6.4609477924811889068410083425e-7, |
| -9.6749773915059089205835337136e-7, |
| 3.6585400766767257736982342461e-7, |
| -8.4592516427275164351876072573e-8, |
| 9.9956786144497936572288988883e-9, |
| 1.4260036420951118112457144842e-9, |
| -1.1761968823382879195380320948e-9, |
| 3.7114575899785204664648987295e-10, |
| -7.4756855194210961661210215325e-11, |
| 7.8536934209183700456512982968e-12, |
| 9.9827182259685539619810406271e-13, |
| -7.5276687030192221587850302453e-13, |
| 2.1955026393964279988917878654e-13, |
| -4.1934859852834647427576319246e-14, |
| 4.6341149635933550715779074274e-15, |
| 2.3742488509048340106830309402e-16, |
| -2.7276516388124786119323824391e-16, |
| 7.8473570134636044722154797225e-17 |
| }; |
| static cheb_series zeta_xgt1_cs = { |
| zeta_xgt1_data, |
| 29, |
| -1, 1, |
| 17 |
| }; |
| |
| |
| /* chebyshev fit for Ln[Zeta[s(t)] - 1 - 2^(-s(t))] |
| * s(t)= 10 + 5t |
| * -1 <= t <= 1; 5 <= s <= 15 |
| */ |
| static double zetam1_inter_data[24] = { |
| -21.7509435653088483422022339374, |
| -5.63036877698121782876372020472, |
| 0.0528041358684229425504861579635, |
| -0.0156381809179670789342700883562, |
| 0.00408218474372355881195080781927, |
| -0.0010264867349474874045036628282, |
| 0.000260469880409886900143834962387, |
| -0.0000676175847209968878098566819447, |
| 0.0000179284472587833525426660171124, |
| -4.83238651318556188834107605116e-6, |
| 1.31913788964999288471371329447e-6, |
| -3.63760500656329972578222188542e-7, |
| 1.01146847513194744989748396574e-7, |
| -2.83215225141806501619105289509e-8, |
| 7.97733710252021423361012829496e-9, |
| -2.25850168553956886676250696891e-9, |
| 6.42269392950164306086395744145e-10, |
| -1.83363861846127284505060843614e-10, |
| 5.25309763895283179960368072104e-11, |
| -1.50958687042589821074710575446e-11, |
| 4.34997545516049244697776942981e-12, |
| -1.25597782748190416118082322061e-12, |
| 3.61280740072222650030134104162e-13, |
| -9.66437239205745207188920348801e-14 |
| }; |
| static cheb_series zetam1_inter_cs = { |
| zetam1_inter_data, |
| 22, |
| -1, 1, |
| 12 |
| }; |
| |
| |
| |
| /* assumes s >= 0 and s != 1.0 */ |
| inline |
| static int |
| riemann_zeta_sgt0(double s, gsl_sf_result * result) |
| { |
| if(s < 1.0) { |
| gsl_sf_result c; |
| cheb_eval_e(&zeta_xlt1_cs, 2.0*s - 1.0, &c); |
| result->val = c.val / (s - 1.0); |
| result->err = c.err / fabs(s-1.0) + GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(s <= 20.0) { |
| double x = (2.0*s - 21.0)/19.0; |
| gsl_sf_result c; |
| cheb_eval_e(&zeta_xgt1_cs, x, &c); |
| result->val = c.val / (s - 1.0); |
| result->err = c.err / (s - 1.0) + GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| double f2 = 1.0 - pow(2.0,-s); |
| double f3 = 1.0 - pow(3.0,-s); |
| double f5 = 1.0 - pow(5.0,-s); |
| double f7 = 1.0 - pow(7.0,-s); |
| result->val = 1.0/(f2*f3*f5*f7); |
| result->err = 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| inline |
| static int |
| riemann_zeta1ms_slt0(double s, gsl_sf_result * result) |
| { |
| if(s > -19.0) { |
| double x = (-19 - 2.0*s)/19.0; |
| gsl_sf_result c; |
| cheb_eval_e(&zeta_xgt1_cs, x, &c); |
| result->val = c.val / (-s); |
| result->err = c.err / (-s) + GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| double f2 = 1.0 - pow(2.0,-(1.0-s)); |
| double f3 = 1.0 - pow(3.0,-(1.0-s)); |
| double f5 = 1.0 - pow(5.0,-(1.0-s)); |
| double f7 = 1.0 - pow(7.0,-(1.0-s)); |
| result->val = 1.0/(f2*f3*f5*f7); |
| result->err = 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| /* works for 5 < s < 15*/ |
| static int |
| riemann_zeta_minus_1_intermediate_s(double s, gsl_sf_result * result) |
| { |
| double t = (s - 10.0)/5.0; |
| gsl_sf_result c; |
| cheb_eval_e(&zetam1_inter_cs, t, &c); |
| result->val = exp(c.val) + pow(2.0, -s); |
| result->err = (c.err + 2.0*GSL_DBL_EPSILON)*result->val; |
| return GSL_SUCCESS; |
| } |
| |
| |
| /* assumes s is large and positive |
| * write: zeta(s) - 1 = zeta(s) * (1 - 1/zeta(s)) |
| * and expand a few terms of the product formula to evaluate 1 - 1/zeta(s) |
| * |
| * works well for s > 15 |
| */ |
| static int |
| riemann_zeta_minus1_large_s(double s, gsl_sf_result * result) |
| { |
| double a = pow( 2.0,-s); |
| double b = pow( 3.0,-s); |
| double c = pow( 5.0,-s); |
| double d = pow( 7.0,-s); |
| double e = pow(11.0,-s); |
| double f = pow(13.0,-s); |
| double t1 = a + b + c + d + e + f; |
| double t2 = a*(b+c+d+e+f) + b*(c+d+e+f) + c*(d+e+f) + d*(e+f) + e*f; |
| /* |
| double t3 = a*(b*(c+d+e+f) + c*(d+e+f) + d*(e+f) + e*f) + |
| b*(c*(d+e+f) + d*(e+f) + e*f) + |
| c*(d*(e+f) + e*f) + |
| d*e*f; |
| double t4 = a*(b*(c*(d + e + f) + d*(e + f) + e*f) + c*(d*(e+f) + e*f) + d*e*f) + |
| b*(c*(d*(e+f) + e*f) + d*e*f) + |
| c*d*e*f; |
| double t5 = b*c*d*e*f + a*c*d*e*f+ a*b*d*e*f+ a*b*c*e*f+ a*b*c*d*f+ a*b*c*d*e; |
| double t6 = a*b*c*d*e*f; |
| */ |
| double numt = t1 - t2 /* + t3 - t4 + t5 - t6 */; |
| double zeta = 1.0/((1.0-a)*(1.0-b)*(1.0-c)*(1.0-d)*(1.0-e)*(1.0-f)); |
| result->val = numt*zeta; |
| result->err = (15.0/s + 1.0) * 6.0*GSL_DBL_EPSILON*result->val; |
| return GSL_SUCCESS; |
| } |
| |
| |
| #if 0 |
| /* zeta(n) */ |
| #define ZETA_POS_TABLE_NMAX 100 |
| static double zeta_pos_int_table_OLD[ZETA_POS_TABLE_NMAX+1] = { |
| -0.50000000000000000000000000000, /* zeta(0) */ |
| 0.0 /* FIXME: DirectedInfinity() */, /* zeta(1) */ |
| 1.64493406684822643647241516665, /* ... */ |
| 1.20205690315959428539973816151, |
| 1.08232323371113819151600369654, |
| 1.03692775514336992633136548646, |
| 1.01734306198444913971451792979, |
| 1.00834927738192282683979754985, |
| 1.00407735619794433937868523851, |
| 1.00200839282608221441785276923, |
| 1.00099457512781808533714595890, |
| 1.00049418860411946455870228253, |
| 1.00024608655330804829863799805, |
| 1.00012271334757848914675183653, |
| 1.00006124813505870482925854511, |
| 1.00003058823630702049355172851, |
| 1.00001528225940865187173257149, |
| 1.00000763719763789976227360029, |
| 1.00000381729326499983985646164, |
| 1.00000190821271655393892565696, |
| 1.00000095396203387279611315204, |
| 1.00000047693298678780646311672, |
| 1.00000023845050272773299000365, |
| 1.00000011921992596531107306779, |
| 1.00000005960818905125947961244, |
| 1.00000002980350351465228018606, |
| 1.00000001490155482836504123466, |
| 1.00000000745071178983542949198, |
| 1.00000000372533402478845705482, |
| 1.00000000186265972351304900640, |
| 1.00000000093132743241966818287, |
| 1.00000000046566290650337840730, |
| 1.00000000023283118336765054920, |
| 1.00000000011641550172700519776, |
| 1.00000000005820772087902700889, |
| 1.00000000002910385044497099687, |
| 1.00000000001455192189104198424, |
| 1.00000000000727595983505748101, |
| 1.00000000000363797954737865119, |
| 1.00000000000181898965030706595, |
| 1.00000000000090949478402638893, |
| 1.00000000000045474737830421540, |
| 1.00000000000022737368458246525, |
| 1.00000000000011368684076802278, |
| 1.00000000000005684341987627586, |
| 1.00000000000002842170976889302, |
| 1.00000000000001421085482803161, |
| 1.00000000000000710542739521085, |
| 1.00000000000000355271369133711, |
| 1.00000000000000177635684357912, |
| 1.00000000000000088817842109308, |
| 1.00000000000000044408921031438, |
| 1.00000000000000022204460507980, |
| 1.00000000000000011102230251411, |
| 1.00000000000000005551115124845, |
| 1.00000000000000002775557562136, |
| 1.00000000000000001387778780973, |
| 1.00000000000000000693889390454, |
| 1.00000000000000000346944695217, |
| 1.00000000000000000173472347605, |
| 1.00000000000000000086736173801, |
| 1.00000000000000000043368086900, |
| 1.00000000000000000021684043450, |
| 1.00000000000000000010842021725, |
| 1.00000000000000000005421010862, |
| 1.00000000000000000002710505431, |
| 1.00000000000000000001355252716, |
| 1.00000000000000000000677626358, |
| 1.00000000000000000000338813179, |
| 1.00000000000000000000169406589, |
| 1.00000000000000000000084703295, |
| 1.00000000000000000000042351647, |
| 1.00000000000000000000021175824, |
| 1.00000000000000000000010587912, |
| 1.00000000000000000000005293956, |
| 1.00000000000000000000002646978, |
| 1.00000000000000000000001323489, |
| 1.00000000000000000000000661744, |
| 1.00000000000000000000000330872, |
| 1.00000000000000000000000165436, |
| 1.00000000000000000000000082718, |
| 1.00000000000000000000000041359, |
| 1.00000000000000000000000020680, |
| 1.00000000000000000000000010340, |
| 1.00000000000000000000000005170, |
| 1.00000000000000000000000002585, |
| 1.00000000000000000000000001292, |
| 1.00000000000000000000000000646, |
| 1.00000000000000000000000000323, |
| 1.00000000000000000000000000162, |
| 1.00000000000000000000000000081, |
| 1.00000000000000000000000000040, |
| 1.00000000000000000000000000020, |
| 1.00000000000000000000000000010, |
| 1.00000000000000000000000000005, |
| 1.00000000000000000000000000003, |
| 1.00000000000000000000000000001, |
| 1.00000000000000000000000000001, |
| 1.00000000000000000000000000000, |
| 1.00000000000000000000000000000, |
| 1.00000000000000000000000000000 |
| }; |
| #endif /* 0 */ |
| |
| |
| /* zeta(n) - 1 */ |
| #define ZETA_POS_TABLE_NMAX 100 |
| static double zetam1_pos_int_table[ZETA_POS_TABLE_NMAX+1] = { |
| -1.5, /* zeta(0) */ |
| 0.0, /* FIXME: Infinity */ /* zeta(1) - 1 */ |
| 0.644934066848226436472415166646, /* zeta(2) - 1 */ |
| 0.202056903159594285399738161511, |
| 0.082323233711138191516003696541, |
| 0.036927755143369926331365486457, |
| 0.017343061984449139714517929790, |
| 0.008349277381922826839797549849, |
| 0.004077356197944339378685238508, |
| 0.002008392826082214417852769232, |
| 0.000994575127818085337145958900, |
| 0.000494188604119464558702282526, |
| 0.000246086553308048298637998047, |
| 0.000122713347578489146751836526, |
| 0.000061248135058704829258545105, |
| 0.000030588236307020493551728510, |
| 0.000015282259408651871732571487, |
| 7.6371976378997622736002935630e-6, |
| 3.8172932649998398564616446219e-6, |
| 1.9082127165539389256569577951e-6, |
| 9.5396203387279611315203868344e-7, |
| 4.7693298678780646311671960437e-7, |
| 2.3845050272773299000364818675e-7, |
| 1.1921992596531107306778871888e-7, |
| 5.9608189051259479612440207935e-8, |
| 2.9803503514652280186063705069e-8, |
| 1.4901554828365041234658506630e-8, |
| 7.4507117898354294919810041706e-9, |
| 3.7253340247884570548192040184e-9, |
| 1.8626597235130490064039099454e-9, |
| 9.3132743241966818287176473502e-10, |
| 4.6566290650337840729892332512e-10, |
| 2.3283118336765054920014559759e-10, |
| 1.1641550172700519775929738354e-10, |
| 5.8207720879027008892436859891e-11, |
| 2.9103850444970996869294252278e-11, |
| 1.4551921891041984235929632245e-11, |
| 7.2759598350574810145208690123e-12, |
| 3.6379795473786511902372363558e-12, |
| 1.8189896503070659475848321007e-12, |
| 9.0949478402638892825331183869e-13, |
| 4.5474737830421540267991120294e-13, |
| 2.2737368458246525152268215779e-13, |
| 1.1368684076802278493491048380e-13, |
| 5.6843419876275856092771829675e-14, |
| 2.8421709768893018554550737049e-14, |
| 1.4210854828031606769834307141e-14, |
| 7.1054273952108527128773544799e-15, |
| 3.5527136913371136732984695340e-15, |
| 1.7763568435791203274733490144e-15, |
| 8.8817842109308159030960913863e-16, |
| 4.4408921031438133641977709402e-16, |
| 2.2204460507980419839993200942e-16, |
| 1.1102230251410661337205445699e-16, |
| 5.5511151248454812437237365905e-17, |
| 2.7755575621361241725816324538e-17, |
| 1.3877787809725232762839094906e-17, |
| 6.9388939045441536974460853262e-18, |
| 3.4694469521659226247442714961e-18, |
| 1.7347234760475765720489729699e-18, |
| 8.6736173801199337283420550673e-19, |
| 4.3368086900206504874970235659e-19, |
| 2.1684043449972197850139101683e-19, |
| 1.0842021724942414063012711165e-19, |
| 5.4210108624566454109187004043e-20, |
| 2.7105054312234688319546213119e-20, |
| 1.3552527156101164581485233996e-20, |
| 6.7762635780451890979952987415e-21, |
| 3.3881317890207968180857031004e-21, |
| 1.6940658945097991654064927471e-21, |
| 8.4703294725469983482469926091e-22, |
| 4.2351647362728333478622704833e-22, |
| 2.1175823681361947318442094398e-22, |
| 1.0587911840680233852265001539e-22, |
| 5.2939559203398703238139123029e-23, |
| 2.6469779601698529611341166842e-23, |
| 1.3234889800848990803094510250e-23, |
| 6.6174449004244040673552453323e-24, |
| 3.3087224502121715889469563843e-24, |
| 1.6543612251060756462299236771e-24, |
| 8.2718061255303444036711056167e-25, |
| 4.1359030627651609260093824555e-25, |
| 2.0679515313825767043959679193e-25, |
| 1.0339757656912870993284095591e-25, |
| 5.1698788284564313204101332166e-26, |
| 2.5849394142282142681277617708e-26, |
| 1.2924697071141066700381126118e-26, |
| 6.4623485355705318034380021611e-27, |
| 3.2311742677852653861348141180e-27, |
| 1.6155871338926325212060114057e-27, |
| 8.0779356694631620331587381863e-28, |
| 4.0389678347315808256222628129e-28, |
| 2.0194839173657903491587626465e-28, |
| 1.0097419586828951533619250700e-28, |
| 5.0487097934144756960847711725e-29, |
| 2.5243548967072378244674341938e-29, |
| 1.2621774483536189043753999660e-29, |
| 6.3108872417680944956826093943e-30, |
| 3.1554436208840472391098412184e-30, |
| 1.5777218104420236166444327830e-30, |
| 7.8886090522101180735205378276e-31 |
| }; |
| |
| |
| #define ZETA_NEG_TABLE_NMAX 99 |
| #define ZETA_NEG_TABLE_SIZE 50 |
| static double zeta_neg_int_table[ZETA_NEG_TABLE_SIZE] = { |
| -0.083333333333333333333333333333, /* zeta(-1) */ |
| 0.008333333333333333333333333333, /* zeta(-3) */ |
| -0.003968253968253968253968253968, /* ... */ |
| 0.004166666666666666666666666667, |
| -0.007575757575757575757575757576, |
| 0.021092796092796092796092796093, |
| -0.083333333333333333333333333333, |
| 0.44325980392156862745098039216, |
| -3.05395433027011974380395433027, |
| 26.4562121212121212121212121212, |
| -281.460144927536231884057971014, |
| 3607.5105463980463980463980464, |
| -54827.583333333333333333333333, |
| 974936.82385057471264367816092, |
| -2.0052695796688078946143462272e+07, |
| 4.7238486772162990196078431373e+08, |
| -1.2635724795916666666666666667e+10, |
| 3.8087931125245368811553022079e+11, |
| -1.2850850499305083333333333333e+13, |
| 4.8241448354850170371581670362e+14, |
| -2.0040310656516252738108421663e+16, |
| 9.1677436031953307756992753623e+17, |
| -4.5979888343656503490437943262e+19, |
| 2.5180471921451095697089023320e+21, |
| -1.5001733492153928733711440151e+23, |
| 9.6899578874635940656497942895e+24, |
| -6.7645882379292820990945242302e+26, |
| 5.0890659468662289689766332916e+28, |
| -4.1147288792557978697665486068e+30, |
| 3.5666582095375556109684574609e+32, |
| -3.3066089876577576725680214670e+34, |
| 3.2715634236478716264211227016e+36, |
| -3.4473782558278053878256455080e+38, |
| 3.8614279832705258893092720200e+40, |
| -4.5892974432454332168863989006e+42, |
| 5.7775386342770431824884825688e+44, |
| -7.6919858759507135167410075972e+46, |
| 1.0813635449971654696354033351e+49, |
| -1.6029364522008965406067102346e+51, |
| 2.5019479041560462843656661499e+53, |
| -4.1067052335810212479752045004e+55, |
| 7.0798774408494580617452972433e+57, |
| -1.2804546887939508790190849756e+60, |
| 2.4267340392333524078020892067e+62, |
| -4.8143218874045769355129570066e+64, |
| 9.9875574175727530680652777408e+66, |
| -2.1645634868435185631335136160e+69, |
| 4.8962327039620553206849224516e+71, /* ... */ |
| -1.1549023923963519663954271692e+74, /* zeta(-97) */ |
| 2.8382249570693706959264156336e+76 /* zeta(-99) */ |
| }; |
| |
| |
| /* coefficients for Maclaurin summation in hzeta() |
| * B_{2j}/(2j)! |
| */ |
| static double hzeta_c[15] = { |
| 1.00000000000000000000000000000, |
| 0.083333333333333333333333333333, |
| -0.00138888888888888888888888888889, |
| 0.000033068783068783068783068783069, |
| -8.2671957671957671957671957672e-07, |
| 2.0876756987868098979210090321e-08, |
| -5.2841901386874931848476822022e-10, |
| 1.3382536530684678832826980975e-11, |
| -3.3896802963225828668301953912e-13, |
| 8.5860620562778445641359054504e-15, |
| -2.1748686985580618730415164239e-16, |
| 5.5090028283602295152026526089e-18, |
| -1.3954464685812523340707686264e-19, |
| 3.5347070396294674716932299778e-21, |
| -8.9535174270375468504026113181e-23 |
| }; |
| |
| #define ETA_POS_TABLE_NMAX 100 |
| static double eta_pos_int_table[ETA_POS_TABLE_NMAX+1] = { |
| 0.50000000000000000000000000000, /* eta(0) */ |
| M_LN2, /* eta(1) */ |
| 0.82246703342411321823620758332, /* ... */ |
| 0.90154267736969571404980362113, |
| 0.94703282949724591757650323447, |
| 0.97211977044690930593565514355, |
| 0.98555109129743510409843924448, |
| 0.99259381992283028267042571313, |
| 0.99623300185264789922728926008, |
| 0.99809429754160533076778303185, |
| 0.99903950759827156563922184570, |
| 0.99951714349806075414409417483, |
| 0.99975768514385819085317967871, |
| 0.99987854276326511549217499282, |
| 0.99993917034597971817095419226, |
| 0.99996955121309923808263293263, |
| 0.99998476421490610644168277496, |
| 0.99999237829204101197693787224, |
| 0.99999618786961011347968922641, |
| 0.99999809350817167510685649297, |
| 0.99999904661158152211505084256, |
| 0.99999952325821554281631666433, |
| 0.99999976161323082254789720494, |
| 0.99999988080131843950322382485, |
| 0.99999994039889239462836140314, |
| 0.99999997019885696283441513311, |
| 0.99999998509923199656878766181, |
| 0.99999999254955048496351585274, |
| 0.99999999627475340010872752767, |
| 0.99999999813736941811218674656, |
| 0.99999999906868228145397862728, |
| 0.99999999953434033145421751469, |
| 0.99999999976716989595149082282, |
| 0.99999999988358485804603047265, |
| 0.99999999994179239904531592388, |
| 0.99999999997089618952980952258, |
| 0.99999999998544809143388476396, |
| 0.99999999999272404460658475006, |
| 0.99999999999636202193316875550, |
| 0.99999999999818101084320873555, |
| 0.99999999999909050538047887809, |
| 0.99999999999954525267653087357, |
| 0.99999999999977262633369589773, |
| 0.99999999999988631316532476488, |
| 0.99999999999994315658215465336, |
| 0.99999999999997157829090808339, |
| 0.99999999999998578914539762720, |
| 0.99999999999999289457268000875, |
| 0.99999999999999644728633373609, |
| 0.99999999999999822364316477861, |
| 0.99999999999999911182158169283, |
| 0.99999999999999955591079061426, |
| 0.99999999999999977795539522974, |
| 0.99999999999999988897769758908, |
| 0.99999999999999994448884878594, |
| 0.99999999999999997224442439010, |
| 0.99999999999999998612221219410, |
| 0.99999999999999999306110609673, |
| 0.99999999999999999653055304826, |
| 0.99999999999999999826527652409, |
| 0.99999999999999999913263826204, |
| 0.99999999999999999956631913101, |
| 0.99999999999999999978315956551, |
| 0.99999999999999999989157978275, |
| 0.99999999999999999994578989138, |
| 0.99999999999999999997289494569, |
| 0.99999999999999999998644747284, |
| 0.99999999999999999999322373642, |
| 0.99999999999999999999661186821, |
| 0.99999999999999999999830593411, |
| 0.99999999999999999999915296705, |
| 0.99999999999999999999957648353, |
| 0.99999999999999999999978824176, |
| 0.99999999999999999999989412088, |
| 0.99999999999999999999994706044, |
| 0.99999999999999999999997353022, |
| 0.99999999999999999999998676511, |
| 0.99999999999999999999999338256, |
| 0.99999999999999999999999669128, |
| 0.99999999999999999999999834564, |
| 0.99999999999999999999999917282, |
| 0.99999999999999999999999958641, |
| 0.99999999999999999999999979320, |
| 0.99999999999999999999999989660, |
| 0.99999999999999999999999994830, |
| 0.99999999999999999999999997415, |
| 0.99999999999999999999999998708, |
| 0.99999999999999999999999999354, |
| 0.99999999999999999999999999677, |
| 0.99999999999999999999999999838, |
| 0.99999999999999999999999999919, |
| 0.99999999999999999999999999960, |
| 0.99999999999999999999999999980, |
| 0.99999999999999999999999999990, |
| 0.99999999999999999999999999995, |
| 0.99999999999999999999999999997, |
| 0.99999999999999999999999999999, |
| 0.99999999999999999999999999999, |
| 1.00000000000000000000000000000, |
| 1.00000000000000000000000000000, |
| 1.00000000000000000000000000000, |
| }; |
| |
| |
| #define ETA_NEG_TABLE_NMAX 99 |
| #define ETA_NEG_TABLE_SIZE 50 |
| static double eta_neg_int_table[ETA_NEG_TABLE_SIZE] = { |
| 0.25000000000000000000000000000, /* eta(-1) */ |
| -0.12500000000000000000000000000, /* eta(-3) */ |
| 0.25000000000000000000000000000, /* ... */ |
| -1.06250000000000000000000000000, |
| 7.75000000000000000000000000000, |
| -86.3750000000000000000000000000, |
| 1365.25000000000000000000000000, |
| -29049.0312500000000000000000000, |
| 800572.750000000000000000000000, |
| -2.7741322625000000000000000000e+7, |
| 1.1805291302500000000000000000e+9, |
| -6.0523980051687500000000000000e+10, |
| 3.6794167785377500000000000000e+12, |
| -2.6170760990658387500000000000e+14, |
| 2.1531418140800295250000000000e+16, |
| -2.0288775575173015930156250000e+18, |
| 2.1708009902623770590275000000e+20, |
| -2.6173826968455814932120125000e+22, |
| 3.5324148876863877826668602500e+24, |
| -5.3042033406864906641493838981e+26, |
| 8.8138218364311576767253114668e+28, |
| -1.6128065107490778547354654864e+31, |
| 3.2355470001722734208527794569e+33, |
| -7.0876727476537493198506645215e+35, |
| 1.6890450341293965779175629389e+38, |
| -4.3639690731216831157655651358e+40, |
| 1.2185998827061261322605065672e+43, |
| -3.6670584803153006180101262324e+45, |
| 1.1859898526302099104271449748e+48, |
| -4.1120769493584015047981746438e+50, |
| 1.5249042436787620309090168687e+53, |
| -6.0349693196941307074572991901e+55, |
| 2.5437161764210695823197691519e+58, |
| -1.1396923802632287851130360170e+61, |
| 5.4180861064753979196802726455e+63, |
| -2.7283654799994373847287197104e+66, |
| 1.4529750514918543238511171663e+69, |
| -8.1705519371067450079777183386e+71, |
| 4.8445781606678367790247757259e+74, |
| -3.0246694206649519336179448018e+77, |
| 1.9858807961690493054169047970e+80, |
| -1.3694474620720086994386818232e+83, |
| 9.9070382984295807826303785989e+85, |
| -7.5103780796592645925968460677e+88, |
| 5.9598418264260880840077992227e+91, |
| -4.9455988887500020399263196307e+94, |
| 4.2873596927020241277675775935e+97, |
| -3.8791952037716162900707994047e+100, |
| 3.6600317773156342245401829308e+103, |
| -3.5978775704117283875784869570e+106 /* eta(-99) */ |
| }; |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| |
| int gsl_sf_hzeta_e(const double s, const double q, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(s <= 1.0 || q <= 0.0) { |
| DOMAIN_ERROR(result); |
| } |
| else { |
| const double max_bits = 54.0; |
| const double ln_term0 = -s * log(q); |
| |
| if(ln_term0 < GSL_LOG_DBL_MIN + 1.0) { |
| UNDERFLOW_ERROR(result); |
| } |
| else if(ln_term0 > GSL_LOG_DBL_MAX - 1.0) { |
| OVERFLOW_ERROR (result); |
| } |
| else if((s > max_bits && q < 1.0) || (s > 0.5*max_bits && q < 0.25)) { |
| result->val = pow(q, -s); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(s > 0.5*max_bits && q < 1.0) { |
| const double p1 = pow(q, -s); |
| const double p2 = pow(q/(1.0+q), s); |
| const double p3 = pow(q/(2.0+q), s); |
| result->val = p1 * (1.0 + p2 + p3); |
| result->err = GSL_DBL_EPSILON * (0.5*s + 2.0) * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| /* Euler-Maclaurin summation formula |
| * [Moshier, p. 400, with several typo corrections] |
| */ |
| const int jmax = 12; |
| const int kmax = 10; |
| int j, k; |
| const double pmax = pow(kmax + q, -s); |
| double scp = s; |
| double pcp = pmax / (kmax + q); |
| double ans = pmax*((kmax+q)/(s-1.0) + 0.5); |
| |
| for(k=0; k<kmax; k++) { |
| ans += pow(k + q, -s); |
| } |
| |
| for(j=0; j<=jmax; j++) { |
| double delta = hzeta_c[j+1] * scp * pcp; |
| ans += delta; |
| if(fabs(delta/ans) < 0.5*GSL_DBL_EPSILON) break; |
| scp *= (s+2*j+1)*(s+2*j+2); |
| pcp /= (kmax + q)*(kmax + q); |
| } |
| |
| result->val = ans; |
| result->err = 2.0 * (jmax + 1.0) * GSL_DBL_EPSILON * fabs(ans); |
| return GSL_SUCCESS; |
| } |
| } |
| } |
| |
| |
| int gsl_sf_zeta_e(const double s, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(s == 1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(s >= 0.0) { |
| return riemann_zeta_sgt0(s, result); |
| } |
| else { |
| /* reflection formula, [Abramowitz+Stegun, 23.2.5] */ |
| |
| gsl_sf_result zeta_one_minus_s; |
| const int stat_zoms = riemann_zeta1ms_slt0(s, &zeta_one_minus_s); |
| const double sin_term = (fmod(s,2.0) == 0.0) ? 0.0 : sin(0.5*M_PI*fmod(s,4.0))/M_PI; |
| |
| if(sin_term == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(s > -170) { |
| /* We have to be careful about losing digits |
| * in calculating pow(2 Pi, s). The gamma |
| * function is fine because we were careful |
| * with that implementation. |
| * We keep an array of (2 Pi)^(10 n). |
| */ |
| const double twopi_pow[18] = { 1.0, |
| 9.589560061550901348e+007, |
| 9.195966217409212684e+015, |
| 8.818527036583869903e+023, |
| 8.456579467173150313e+031, |
| 8.109487671573504384e+039, |
| 7.776641909496069036e+047, |
| 7.457457466828644277e+055, |
| 7.151373628461452286e+063, |
| 6.857852693272229709e+071, |
| 6.576379029540265771e+079, |
| 6.306458169130020789e+087, |
| 6.047615938853066678e+095, |
| 5.799397627482402614e+103, |
| 5.561367186955830005e+111, |
| 5.333106466365131227e+119, |
| 5.114214477385391780e+127, |
| 4.904306689854036836e+135 |
| }; |
| const int n = floor((-s)/10.0); |
| const double fs = s + 10.0*n; |
| const double p = pow(2.0*M_PI, fs) / twopi_pow[n]; |
| |
| gsl_sf_result g; |
| const int stat_g = gsl_sf_gamma_e(1.0-s, &g); |
| result->val = p * g.val * sin_term * zeta_one_minus_s.val; |
| result->err = fabs(p * g.val * sin_term) * zeta_one_minus_s.err; |
| result->err += fabs(p * sin_term * zeta_one_minus_s.val) * g.err; |
| result->err += GSL_DBL_EPSILON * (fabs(s)+2.0) * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_g, stat_zoms); |
| } |
| else { |
| /* The actual zeta function may or may not |
| * overflow here. But we have no easy way |
| * to calculate it when the prefactor(s) |
| * overflow. Trying to use log's and exp |
| * is no good because we loose a couple |
| * digits to the exp error amplification. |
| * When we gather a little more patience, |
| * we can implement something here. Until |
| * then just give up. |
| */ |
| OVERFLOW_ERROR(result); |
| } |
| } |
| } |
| |
| |
| int gsl_sf_zeta_int_e(const int n, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(n < 0) { |
| if(!GSL_IS_ODD(n)) { |
| result->val = 0.0; /* exactly zero at even negative integers */ |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(n > -ZETA_NEG_TABLE_NMAX) { |
| result->val = zeta_neg_int_table[-(n+1)/2]; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| return gsl_sf_zeta_e((double)n, result); |
| } |
| } |
| else if(n == 1){ |
| DOMAIN_ERROR(result); |
| } |
| else if(n <= ZETA_POS_TABLE_NMAX){ |
| result->val = 1.0 + zetam1_pos_int_table[n]; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| result->val = 1.0; |
| result->err = GSL_DBL_EPSILON; |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int gsl_sf_zetam1_e(const double s, gsl_sf_result * result) |
| { |
| if(s <= 5.0) |
| { |
| int stat = gsl_sf_zeta_e(s, result); |
| result->val = result->val - 1.0; |
| return stat; |
| } |
| else if(s < 15.0) |
| { |
| return riemann_zeta_minus_1_intermediate_s(s, result); |
| } |
| else |
| { |
| return riemann_zeta_minus1_large_s(s, result); |
| } |
| } |
| |
| |
| int gsl_sf_zetam1_int_e(const int n, gsl_sf_result * result) |
| { |
| if(n < 0) { |
| if(!GSL_IS_ODD(n)) { |
| result->val = -1.0; /* at even negative integers zetam1 == -1 since zeta is exactly zero */ |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(n > -ZETA_NEG_TABLE_NMAX) { |
| result->val = zeta_neg_int_table[-(n+1)/2] - 1.0; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| /* could use gsl_sf_zetam1_e here but subtracting 1 makes no difference |
| for such large values, so go straight to the result */ |
| return gsl_sf_zeta_e((double)n, result); |
| } |
| } |
| else if(n == 1){ |
| DOMAIN_ERROR(result); |
| } |
| else if(n <= ZETA_POS_TABLE_NMAX){ |
| result->val = zetam1_pos_int_table[n]; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| return gsl_sf_zetam1_e(n, result); |
| } |
| } |
| |
| |
| int gsl_sf_eta_int_e(int n, gsl_sf_result * result) |
| { |
| if(n > ETA_POS_TABLE_NMAX) { |
| result->val = 1.0; |
| result->err = GSL_DBL_EPSILON; |
| return GSL_SUCCESS; |
| } |
| else if(n >= 0) { |
| result->val = eta_pos_int_table[n]; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| /* n < 0 */ |
| |
| if(!GSL_IS_ODD(n)) { |
| /* exactly zero at even negative integers */ |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(n > -ETA_NEG_TABLE_NMAX) { |
| result->val = eta_neg_int_table[-(n+1)/2]; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| gsl_sf_result z; |
| gsl_sf_result p; |
| int stat_z = gsl_sf_zeta_int_e(n, &z); |
| int stat_p = gsl_sf_exp_e((1.0-n)*M_LN2, &p); |
| int stat_m = gsl_sf_multiply_e(-p.val, z.val, result); |
| result->err = fabs(p.err * (M_LN2*(1.0-n)) * z.val) + z.err * fabs(p.val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_3(stat_m, stat_p, stat_z); |
| } |
| } |
| } |
| |
| |
| int gsl_sf_eta_e(const double s, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(s > 100.0) { |
| result->val = 1.0; |
| result->err = GSL_DBL_EPSILON; |
| return GSL_SUCCESS; |
| } |
| else if(fabs(s-1.0) < 10.0*GSL_ROOT5_DBL_EPSILON) { |
| double del = s-1.0; |
| double c0 = M_LN2; |
| double c1 = M_LN2 * (M_EULER - 0.5*M_LN2); |
| double c2 = -0.0326862962794492996; |
| double c3 = 0.0015689917054155150; |
| double c4 = 0.00074987242112047532; |
| result->val = c0 + del * (c1 + del * (c2 + del * (c3 + del * c4))); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| gsl_sf_result z; |
| gsl_sf_result p; |
| int stat_z = gsl_sf_zeta_e(s, &z); |
| int stat_p = gsl_sf_exp_e((1.0-s)*M_LN2, &p); |
| int stat_m = gsl_sf_multiply_e(1.0-p.val, z.val, result); |
| result->err = fabs(p.err * (M_LN2*(1.0-s)) * z.val) + z.err * fabs(p.val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_3(stat_m, stat_p, stat_z); |
| } |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_zeta(const double s) |
| { |
| EVAL_RESULT(gsl_sf_zeta_e(s, &result)); |
| } |
| |
| double gsl_sf_hzeta(const double s, const double a) |
| { |
| EVAL_RESULT(gsl_sf_hzeta_e(s, a, &result)); |
| } |
| |
| double gsl_sf_zeta_int(const int s) |
| { |
| EVAL_RESULT(gsl_sf_zeta_int_e(s, &result)); |
| } |
| |
| double gsl_sf_zetam1(const double s) |
| { |
| EVAL_RESULT(gsl_sf_zetam1_e(s, &result)); |
| } |
| |
| double gsl_sf_zetam1_int(const int s) |
| { |
| EVAL_RESULT(gsl_sf_zetam1_int_e(s, &result)); |
| } |
| |
| double gsl_sf_eta_int(const int s) |
| { |
| EVAL_RESULT(gsl_sf_eta_int_e(s, &result)); |
| } |
| |
| double gsl_sf_eta(const double s) |
| { |
| EVAL_RESULT(gsl_sf_eta_e(s, &result)); |
| } |