psi.c

开放gsl矩阵运算

/* specfunc/psi.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_exp.h"
#include "gsl_sf_gamma.h"
#include "gsl_sf_zeta.h"
#include "gsl_sf_psi.h"

#include "error.h"

#include "chebyshev.h"
#include "cheb_eval.c"

/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/


/* Chebyshev fit for f(y) = Re(Psi(1+Iy)) + M_EULER - y^2/(1+y^2) - y^2/(2(4+y^2))
 * 1 < y < 10
 *   ==>
 * y(x) = (9x + 11)/2,  -1 < x < 1
 * x(y) = (2y - 11)/9
 *
 * g(x) := f(y(x))
 */
static double r1py_data[] = {
   1.59888328244976954803168395603,
   0.67905625353213463845115658455,
  -0.068485802980122530009506482524,
  -0.005788184183095866792008831182,
   0.008511258167108615980419855648,
  -0.004042656134699693434334556409,
   0.001352328406159402601778462956,
  -0.000311646563930660566674525382,
   0.000018507563785249135437219139,
   0.000028348705427529850296492146,
  -0.000019487536014574535567541960,
   8.0709788710834469408621587335e-06,
  -2.2983564321340518037060346561e-06,
   3.0506629599604749843855962658e-07,
   1.3042238632418364610774284846e-07,
  -1.2308657181048950589464690208e-07,
   5.7710855710682427240667414345e-08,
  -1.8275559342450963966092636354e-08,
   3.1020471300626589420759518930e-09,
   6.8989327480593812470039430640e-10,
  -8.7182290258923059852334818997e-10,
   4.4069147710243611798213548777e-10,
  -1.4727311099198535963467200277e-10,
   2.7589682523262644748825844248e-11,
   4.1871826756975856411554363568e-12,
  -6.5673460487260087541400767340e-12,
   3.4487900886723214020103638000e-12,
  -1.1807251417448690607973794078e-12,
   2.3798314343969589258709315574e-13,
   2.1663630410818831824259465821e-15
};
static cheb_series r1py_cs = {
  r1py_data,
  29,
  -1,1,
  18
};


/* Chebyshev fits from SLATEC code for psi(x)

 Series for PSI        on the interval  0.	   to  1.00000D+00
				       with weighted error   2.03E-17
					log weighted error  16.69
			      significant figures required  16.39
				   decimal places required  17.37

 Series for APSI       on the interval  0.	   to  2.50000D-01
				       with weighted error   5.54E-17
					log weighted error  16.26
			      significant figures required  14.42
				   decimal places required  16.86

*/

static double psics_data[23] = {
  -.038057080835217922,
   .491415393029387130, 
  -.056815747821244730,
   .008357821225914313,
  -.001333232857994342,
   .000220313287069308,
  -.000037040238178456,
   .000006283793654854,
  -.000001071263908506,
   .000000183128394654,
  -.000000031353509361,
   .000000005372808776,
  -.000000000921168141,
   .000000000157981265,
  -.000000000027098646,
   .000000000004648722,
  -.000000000000797527,
   .000000000000136827,
  -.000000000000023475,
   .000000000000004027,
  -.000000000000000691,
   .000000000000000118,
  -.000000000000000020
};
static cheb_series psi_cs = {
  psics_data,
  22,
  -1, 1,
  17
};

static double apsics_data[16] = {    
  -.0204749044678185,
  -.0101801271534859,
   .0000559718725387,
  -.0000012917176570,
   .0000000572858606,
  -.0000000038213539,
   .0000000003397434,
  -.0000000000374838,
   .0000000000048990,
  -.0000000000007344,
   .0000000000001233,
  -.0000000000000228,
   .0000000000000045,
  -.0000000000000009,
   .0000000000000002,
  -.0000000000000000 
};    
static cheb_series apsi_cs = {
  apsics_data,
  15,
  -1, 1,
  9
};

#define PSI_TABLE_NMAX 100
static double psi_table[PSI_TABLE_NMAX+1] = {
  0.0,  /* Infinity */              /* psi(0) */
 -M_EULER,                          /* psi(1) */
  0.42278433509846713939348790992,  /* ...    */
  0.92278433509846713939348790992,
  1.25611766843180047272682124325,
  1.50611766843180047272682124325,
  1.70611766843180047272682124325,
  1.87278433509846713939348790992,
  2.01564147795560999653634505277,
  2.14064147795560999653634505277,
  2.25175258906672110764745616389,
  2.35175258906672110764745616389,
  2.44266167997581201673836525479,
  2.52599501330914535007169858813,
  2.60291809023222227314862166505,
  2.67434666166079370172005023648,
  2.74101332832746036838671690315,
  2.80351332832746036838671690315,
  2.86233685773922507426906984432,
  2.91789241329478062982462539988,
  2.97052399224214905087725697883,
  3.02052399224214905087725697883,
  3.06814303986119666992487602645,
  3.11359758531574212447033057190,
  3.15707584618530734186163491973,
  3.1987425128519740085283015864,
  3.2387425128519740085283015864,
  3.2772040513135124700667631249,
  3.3142410883505495071038001619,
  3.3499553740648352213895144476,
  3.3844381326855248765619282407,
  3.4177714660188582098952615740,
  3.4500295305349872421533260902,
  3.4812795305349872421533260902,
  3.5115825608380175451836291205,
  3.5409943255438998981248055911,
  3.5695657541153284695533770196,
  3.5973435318931062473311547974,
  3.6243705589201332743581818244,
  3.6506863483938174848844976139,
  3.6763273740348431259101386396,
  3.7013273740348431259101386396,
  3.7257176179372821503003825420,
  3.7495271417468059598241920658,
  3.7727829557002943319172153216,
  3.7955102284275670591899425943,
  3.8177324506497892814121648166,
  3.8394715810845718901078169905,
  3.8607481768292527411716467777,
  3.8815815101625860745049801110,
  3.9019896734278921969539597029,
  3.9219896734278921969539597029,
  3.9415975165651470989147440166,
  3.9608282857959163296839747858,
  3.9796962103242182164764276160,
  3.9982147288427367349949461345,
  4.0163965470245549168131279527,
  4.0342536898816977739559850956,
  4.0517975495308205809735289552,
  4.0690389288411654085597358518,
  4.0859880813835382899156680552,
  4.1026547480502049565823347218,
  4.1190481906731557762544658694,
  4.1351772229312202923834981274,
  4.1510502388042361653993711433,
  4.1666752388042361653993711433,
  4.1820598541888515500147557587,
  4.1972113693403667015299072739,
  4.2121367424746950597388624977,
  4.2268426248276362362094507330,
  4.2413353784508246420065521823,
  4.2556210927365389277208378966,
  4.2697055997787924488475984600,
  4.2835944886676813377364873489,
  4.2972931188046676391063503626,
  4.3108066323181811526198638761,
  4.3241399656515144859531972094,
  4.3372978603883565912163551041,
  4.3502848733753695782293421171,
  4.3631053861958823987421626300,
  4.3757636140439836645649474401,
  4.3882636140439836645649474401,
  4.4006092930563293435772931191,
  4.4128044150075488557724150703,
  4.4248526077786331931218126607,
  4.4367573696833950978837174226,
  4.4485220755657480390601880108,
  4.4601499825424922251066996387,
  4.4716442354160554434975042364,
  4.4830078717796918071338678728,
  4.4942438268358715824147667492,
  4.5053549379469826935258778603,
  4.5163439489359936825368668713,
  4.5272135141533849868846929582,
  4.5379662023254279976373811303,
  4.5486045001977684231692960239,
  4.5591308159872421073798223397,
  4.5695474826539087740464890064,
  4.5798567610044242379640147796,
  4.5900608426370772991885045755,
  4.6001618527380874001986055856
};


#define PSI_1_TABLE_NMAX 100
static double psi_1_table[PSI_1_TABLE_NMAX+1] = {
  0.0,  /* Infinity */              /* psi(1,0) */
 -M_PI*M_PI/6.0,                    /* psi(1,1) */
  0.644934066848226436472415,       /* ...      */
  0.394934066848226436472415,
  0.2838229557371153253613041,
  0.2213229557371153253613041,
  0.1813229557371153253613041,
  0.1535451779593375475835263,
  0.1331370146940314251345467,
  0.1175120146940314251345467,
  0.1051663356816857461222010,
  0.0951663356816857461222010,
  0.0869018728717683907503002,
  0.0799574284273239463058557,
  0.0740402686640103368384001,
  0.0689382278476838062261552,
  0.0644937834032393617817108,
  0.0605875334032393617817108,
  0.0571273257907826143768665,
  0.0540409060376961946237801,
  0.0512708229352031198315363,
  0.0487708229352031198315363,
  0.0465032492390579951149830,
  0.0444371335365786562720078,
  0.0425467743683366902984728,
  0.0408106632572255791873617,
  0.0392106632572255791873617,
  0.0377313733163971768204978,
  0.0363596312039143235969038,
  0.0350841209998326909438426,
  0.0338950603577399442137594,
  0.0327839492466288331026483,
  0.0317433665203020901265817,
  0.03076680402030209012658168,
  0.02984853037475571730748159,
  0.02898347847164153045627052,
  0.02816715194102928555831133,
  0.02739554700275768062003973,
  0.02666508681283803124093089,
  0.02597256603721476254286995,
  0.02531510384129102815759710,
  0.02469010384129102815759710,
  0.02409521984367056414807896,
  0.02352832641963428296894063,
  0.02298749353699501850166102,
  0.02247096461137518379091722,
  0.02197713745088135663042339,
  0.02150454765882086513703965,
  0.02105185413233829383780923,
  0.02061782635456051606003145,
  0.02020133322669712580597065,
  0.01980133322669712580597065,
  0.01941686571420193164987683,