/* 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 3 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));
}