zeta.c

开放gsl矩阵运算

/* specfunc/zeta.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
 * 
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or (at
 * your option) any later version.
 * 
 * This program is distributed in the hope that it will be useful, but
 * WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * General Public License for more details.
 * 
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
 */

/* Author:  G. Jungman */

#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include "gsl_sf_elementary.h"
#include "gsl_sf_exp.h"
#include "gsl_sf_gamma.h"
#include "gsl_sf_pow_int.h"
#include "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
};


/* 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_zeta1m_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;
  }
}

/* zeta(n) */
#define ZETA_POS_TABLE_NMAX   100
static double zeta_pos_int_table[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
};
#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,