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-rw-r--r--gsl-1.9/specfunc/zeta.c1050
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diff --git a/gsl-1.9/specfunc/zeta.c b/gsl-1.9/specfunc/zeta.c
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+++ b/gsl-1.9/specfunc/zeta.c
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+/* 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));
+}
generated by cgit v1.2.3 (git 2.25.1) at 2024-05-01 06:40:28 +0000