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- /* specfunc/psi.c
- *
- * Copyright (C) 2007 Brian Gough
- * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004, 2005, 2006 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 "gsl__config.h"
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "gsl_sf_exp.h"
- #include "gsl_sf_gamma.h"
- #include "gsl_sf_zeta.h"
- #include "gsl_sf_psi.h"
- #include "gsl_complex_math.h"
- #include <stdio.h>
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__chebyshev.h"
- #include "gsl_specfunc__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,
- 0.01904704322899483105816086,
- 0.01869104465298913508094477,
- 0.01834810912486842177504628,
- 0.01801753061247172756017024,
- 0.01769865306145131939690494,
- 0.01739086605006319997554452,
- 0.01709360088954001329302371,
- 0.01680632711763538818529605,
- 0.01652854933985761040751827,
- 0.01625980437882562975715546,
- 0.01599965869724394401313881,
- 0.01574770606433893015574400,
- 0.01550356543933893015574400,
- 0.01526687904880638577704578,
- 0.01503731063741979257227076,
- 0.01481454387422086185273411,
- 0.01459828089844231513993134,
- 0.01438824099085987447620523,
- 0.01418415935820681325171544,
- 0.01398578601958352422176106,
- 0.01379288478501562298719316,
- 0.01360523231738567365335942,
- 0.01342261726990576130858221,
- 0.01324483949212798353080444,
- 0.01307170929822216635628920,
- 0.01290304679189732236910755,
- 0.01273868124291638877278934,
- 0.01257845051066194236996928,
- 0.01242220051066194236996928,
- 0.01226978472038606978956995,
- 0.01212106372098095378719041,
- 0.01197590477193174490346273,
- 0.01183418141592267460867815,
- 0.01169577311142440471248438,
- 0.01156056489076458859566448,
- 0.01142844704164317229232189,
- 0.01129931481023821361463594,
- 0.01117306812421372175754719,
- 0.01104961133409026496742374,
- 0.01092885297157366069257770,
- 0.01081070552355853781923177,
- 0.01069508522063334415522437,
- 0.01058191183901270133041676,
- 0.01047110851491297833872701,
- 0.01036260157046853389428257,
- 0.01025632035036012704977199, /* ... */
- 0.01015219706839427948625679, /* psi(1,99) */
- 0.01005016666333357139524567 /* psi(1,100) */
- };
- /* digamma for x both positive and negative; we do both
- * cases here because of the way we use even/odd parts
- * of the function
- */
- static int
- psi_x(const double x, gsl_sf_result * result)
- {
- const double y = fabs(x);
- if(x == 0.0 || x == -1.0 || x == -2.0) {
- DOMAIN_ERROR(result);
- }
- else if(y >= 2.0) {
- const double t = 8.0/(y*y)-1.0;
- gsl_sf_result result_c;
- cheb_eval_e(&apsi_cs, t, &result_c);
- if(x < 0.0) {
- const double s = sin(M_PI*x);
- const double c = cos(M_PI*x);
- if(fabs(s) < 2.0*GSL_SQRT_DBL_MIN) {
- DOMAIN_ERROR(result);
- }
- else {
- result->val = log(y) - 0.5/x + result_c.val - M_PI * c/s;
- result->err = M_PI*fabs(x)*GSL_DBL_EPSILON/(s*s);
- result->err += result_c.err;
- result->err += GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- else {
- result->val = log(y) - 0.5/x + result_c.val;
- result->err = result_c.err;
- result->err += GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- else { /* -2 < x < 2 */
- gsl_sf_result result_c;
- if(x < -1.0) { /* x = -2 + v */
- const double v = x + 2.0;
- const double t1 = 1.0/x;
- const double t2 = 1.0/(x+1.0);
- const double t3 = 1.0/v;
- cheb_eval_e(&psi_cs, 2.0*v-1.0, &result_c);
-
- result->val = -(t1 + t2 + t3) + result_c.val;
- result->err = GSL_DBL_EPSILON * (fabs(t1) + fabs(x/(t2*t2)) + fabs(x/(t3*t3)));
- result->err += result_c.err;
- result->err += GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else if(x < 0.0) { /* x = -1 + v */
- const double v = x + 1.0;
- const double t1 = 1.0/x;
- const double t2 = 1.0/v;
- cheb_eval_e(&psi_cs, 2.0*v-1.0, &result_c);
-
- result->val = -(t1 + t2) + result_c.val;
- result->err = GSL_DBL_EPSILON * (fabs(t1) + fabs(x/(t2*t2)));
- result->err += result_c.err;
- result->err += GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else if(x < 1.0) { /* x = v */
- const double t1 = 1.0/x;
- cheb_eval_e(&psi_cs, 2.0*x-1.0, &result_c);
-
- result->val = -t1 + result_c.val;
- result->err = GSL_DBL_EPSILON * t1;
- result->err += result_c.err;
- result->err += GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else { /* x = 1 + v */
- const double v = x - 1.0;
- return cheb_eval_e(&psi_cs, 2.0*v-1.0, result);
- }
- }
- }
- /* psi(z) for large |z| in the right half-plane; [Abramowitz + Stegun, 6.3.18] */
- static
- gsl_complex
- psi_complex_asymp(gsl_complex z)
- {
- /* coefficients in the asymptotic expansion for large z;
- * let w = z^(-2) and write the expression in the form
- *
- * ln(z) - 1/(2z) - 1/12 w (1 + c1 w + c2 w + c3 w + ... )
- */
- static const double c1 = -0.1;
- static const double c2 = 1.0/21.0;
- static const double c3 = -0.05;
- gsl_complex zi = gsl_complex_inverse(z);
- gsl_complex w = gsl_complex_mul(zi, zi);
- gsl_complex cs;
- /* Horner method evaluation of term in parentheses */
- gsl_complex sum;
- sum = gsl_complex_mul_real(w, c3/c2);
- sum = gsl_complex_add_real(sum, 1.0);
- sum = gsl_complex_mul_real(sum, c2/c1);
- sum = gsl_complex_mul(sum, w);
- sum = gsl_complex_add_real(sum, 1.0);
- sum = gsl_complex_mul_real(sum, c1);
- sum = gsl_complex_mul(sum, w);
- sum = gsl_complex_add_real(sum, 1.0);
- /* correction added to log(z) */
- cs = gsl_complex_mul(sum, w);
- cs = gsl_complex_mul_real(cs, -1.0/12.0);
- cs = gsl_complex_add(cs, gsl_complex_mul_real(zi, -0.5));
- return gsl_complex_add(gsl_complex_log(z), cs);
- }
- /* psi(z) for complex z in the right half-plane */
- static int
- psi_complex_rhp(
- gsl_complex z,
- gsl_sf_result * result_re,
- gsl_sf_result * result_im
- )
- {
- int n_recurse = 0;
- int i;
- gsl_complex a;
- if(GSL_REAL(z) == 0.0 && GSL_IMAG(z) == 0.0)
- {
- result_re->val = 0.0;
- result_im->val = 0.0;
- result_re->err = 0.0;
- result_im->err = 0.0;
- return GSL_EDOM;
- }
- /* compute the number of recurrences to apply */
- if(GSL_REAL(z) < 20.0 && fabs(GSL_IMAG(z)) < 20.0)
- {
- const double sp = sqrt(20.0 + GSL_IMAG(z));
- const double sn = sqrt(20.0 - GSL_IMAG(z));
- const double rhs = sp*sn - GSL_REAL(z);
- if(rhs > 0.0) n_recurse = ceil(rhs);
- }
- /* compute asymptotic at the large value z + n_recurse */
- a = psi_complex_asymp(gsl_complex_add_real(z, n_recurse));
- result_re->err = 2.0 * GSL_DBL_EPSILON * fabs(GSL_REAL(a));
- result_im->err = 2.0 * GSL_DBL_EPSILON * fabs(GSL_IMAG(a));
- /* descend recursively, if necessary */
- for(i = n_recurse; i >= 1; --i)
- {
- gsl_complex zn = gsl_complex_add_real(z, i - 1.0);
- gsl_complex zn_inverse = gsl_complex_inverse(zn);
- a = gsl_complex_sub(a, zn_inverse);
- /* accumulate the error, to catch cancellations */
- result_re->err += 2.0 * GSL_DBL_EPSILON * fabs(GSL_REAL(zn_inverse));
- result_im->err += 2.0 * GSL_DBL_EPSILON * fabs(GSL_IMAG(zn_inverse));
- }
- result_re->val = GSL_REAL(a);
- result_im->val = GSL_IMAG(a);
- result_re->err += 2.0 * GSL_DBL_EPSILON * fabs(result_re->val);
- result_im->err += 2.0 * GSL_DBL_EPSILON * fabs(result_im->val);
- return GSL_SUCCESS;
- }
- /* generic polygamma; assumes n >= 0 and x > 0
- */
- static int
- psi_n_xg0(const int n, const double x, gsl_sf_result * result)
- {
- if(n == 0) {
- return gsl_sf_psi_e(x, result);
- }
- else {
- /* Abramowitz + Stegun 6.4.10 */
- gsl_sf_result ln_nf;
- gsl_sf_result hzeta;
- int stat_hz = gsl_sf_hzeta_e(n+1.0, x, &hzeta);
- int stat_nf = gsl_sf_lnfact_e((unsigned int) n, &ln_nf);
- int stat_e = gsl_sf_exp_mult_err_e(ln_nf.val, ln_nf.err,
- hzeta.val, hzeta.err,
- result);
- if(GSL_IS_EVEN(n)) result->val = -result->val;
- return GSL_ERROR_SELECT_3(stat_e, stat_nf, stat_hz);
- }
- }
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
- int gsl_sf_psi_int_e(const int n, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(n <= 0) {
- DOMAIN_ERROR(result);
- }
- else if(n <= PSI_TABLE_NMAX) {
- result->val = psi_table[n];
- result->err = GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- /* Abramowitz+Stegun 6.3.18 */
- const double c2 = -1.0/12.0;
- const double c3 = 1.0/120.0;
- const double c4 = -1.0/252.0;
- const double c5 = 1.0/240.0;
- const double ni2 = (1.0/n)*(1.0/n);
- const double ser = ni2 * (c2 + ni2 * (c3 + ni2 * (c4 + ni2*c5)));
- result->val = log(n) - 0.5/n + ser;
- result->err = GSL_DBL_EPSILON * (fabs(log(n)) + fabs(0.5/n) + fabs(ser));
- result->err += GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_psi_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- return psi_x(x, result);
- }
- int
- gsl_sf_psi_1piy_e(const double y, gsl_sf_result * result)
- {
- const double ay = fabs(y);
- /* CHECK_POINTER(result) */
- if(ay > 1000.0) {
- /* [Abramowitz+Stegun, 6.3.19] */
- const double yi2 = 1.0/(ay*ay);
- const double lny = log(ay);
- const double sum = yi2 * (1.0/12.0 + 1.0/120.0 * yi2 + 1.0/252.0 * yi2*yi2);
- result->val = lny + sum;
- result->err = 2.0 * GSL_DBL_EPSILON * (fabs(lny) + fabs(sum));
- return GSL_SUCCESS;
- }
- else if(ay > 10.0) {
- /* [Abramowitz+Stegun, 6.3.19] */
- const double yi2 = 1.0/(ay*ay);
- const double lny = log(ay);
- const double sum = yi2 * (1.0/12.0 +
- yi2 * (1.0/120.0 +
- yi2 * (1.0/252.0 +
- yi2 * (1.0/240.0 +
- yi2 * (1.0/132.0 + 691.0/32760.0 * yi2)))));
- result->val = lny + sum;
- result->err = 2.0 * GSL_DBL_EPSILON * (fabs(lny) + fabs(sum));
- return GSL_SUCCESS;
- }
- else if(ay > 1.0){
- const double y2 = ay*ay;
- const double x = (2.0*ay - 11.0)/9.0;
- const double v = y2*(1.0/(1.0+y2) + 0.5/(4.0+y2));
- gsl_sf_result result_c;
- cheb_eval_e(&r1py_cs, x, &result_c);
- result->val = result_c.val - M_EULER + v;
- result->err = result_c.err;
- result->err += 2.0 * GSL_DBL_EPSILON * (fabs(v) + M_EULER + fabs(result_c.val));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- result->err *= 5.0; /* FIXME: losing a digit somewhere... maybe at x=... ? */
- return GSL_SUCCESS;
- }
- else {
- /* [Abramowitz+Stegun, 6.3.17]
- *
- * -M_EULER + y^2 Sum[1/n 1/(n^2 + y^2), {n,1,M}]
- * + Sum[1/n^3, {n,M+1,Infinity}]
- * - y^2 Sum[1/n^5, {n,M+1,Infinity}]
- * + y^4 Sum[1/n^7, {n,M+1,Infinity}]
- * - y^6 Sum[1/n^9, {n,M+1,Infinity}]
- * + O(y^8)
- *
- * We take M=50 for at least 15 digit precision.
- */
- const int M = 50;
- const double y2 = y*y;
- const double c0 = 0.00019603999466879846570;
- const double c2 = 3.8426659205114376860e-08;
- const double c4 = 1.0041592839497643554e-11;
- const double c6 = 2.9516743763500191289e-15;
- const double p = c0 + y2 *(-c2 + y2*(c4 - y2*c6));
- double sum = 0.0;
- double v;
-
- int n;
- for(n=1; n<=M; n++) {
- sum += 1.0/(n * (n*n + y*y));
- }
- v = y2 * (sum + p);
- result->val = -M_EULER + v;
- result->err = GSL_DBL_EPSILON * (M_EULER + fabs(v));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_psi_1_int_e(const int n, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(n <= 0) {
- DOMAIN_ERROR(result);
- }
- else if(n <= PSI_1_TABLE_NMAX) {
- result->val = psi_1_table[n];
- result->err = GSL_DBL_EPSILON * result->val;
- return GSL_SUCCESS;
- }
- else {
- /* Abramowitz+Stegun 6.4.12
- * double-precision for n > 100
- */
- const double c0 = -1.0/30.0;
- const double c1 = 1.0/42.0;
- const double c2 = -1.0/30.0;
- const double ni2 = (1.0/n)*(1.0/n);
- const double ser = ni2*ni2 * (c0 + ni2*(c1 + c2*ni2));
- result->val = (1.0 + 0.5/n + 1.0/(6.0*n*n) + ser) / n;
- result->err = GSL_DBL_EPSILON * result->val;
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_psi_1_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x == 0.0 || x == -1.0 || x == -2.0) {
- DOMAIN_ERROR(result);
- }
- else if(x > 0.0)
- {
- return psi_n_xg0(1, x, result);
- }
- else if(x > -5.0)
- {
- /* Abramowitz + Stegun 6.4.6 */
- int M = -floor(x);
- double fx = x + M;
- double sum = 0.0;
- int m;
- if(fx == 0.0)
- DOMAIN_ERROR(result);
- for(m = 0; m < M; ++m)
- sum += 1.0/((x+m)*(x+m));
- {
- int stat_psi = psi_n_xg0(1, fx, result);
- result->val += sum;
- result->err += M * GSL_DBL_EPSILON * sum;
- return stat_psi;
- }
- }
- else
- {
- /* Abramowitz + Stegun 6.4.7 */
- const double sin_px = sin(M_PI * x);
- const double d = M_PI*M_PI/(sin_px*sin_px);
- gsl_sf_result r;
- int stat_psi = psi_n_xg0(1, 1.0-x, &r);
- result->val = d - r.val;
- result->err = r.err + 2.0*GSL_DBL_EPSILON*d;
- return stat_psi;
- }
- }
- int gsl_sf_psi_n_e(const int n, const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(n == 0)
- {
- return gsl_sf_psi_e(x, result);
- }
- else if(n == 1)
- {
- return gsl_sf_psi_1_e(x, result);
- }
- else if(n < 0 || x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else {
- gsl_sf_result ln_nf;
- gsl_sf_result hzeta;
- int stat_hz = gsl_sf_hzeta_e(n+1.0, x, &hzeta);
- int stat_nf = gsl_sf_lnfact_e((unsigned int) n, &ln_nf);
- int stat_e = gsl_sf_exp_mult_err_e(ln_nf.val, ln_nf.err,
- hzeta.val, hzeta.err,
- result);
- if(GSL_IS_EVEN(n)) result->val = -result->val;
- return GSL_ERROR_SELECT_3(stat_e, stat_nf, stat_hz);
- }
- }
- int
- gsl_sf_complex_psi_e(
- const double x,
- const double y,
- gsl_sf_result * result_re,
- gsl_sf_result * result_im
- )
- {
- if(x >= 0.0)
- {
- gsl_complex z = gsl_complex_rect(x, y);
- return psi_complex_rhp(z, result_re, result_im);
- }
- else
- {
- /* reflection formula [Abramowitz+Stegun, 6.3.7] */
- gsl_complex z = gsl_complex_rect(x, y);
- gsl_complex omz = gsl_complex_rect(1.0 - x, -y);
- gsl_complex zpi = gsl_complex_mul_real(z, M_PI);
- gsl_complex cotzpi = gsl_complex_cot(zpi);
- int ret_val = psi_complex_rhp(omz, result_re, result_im);
- if(GSL_IS_REAL(GSL_REAL(cotzpi)) && GSL_IS_REAL(GSL_IMAG(cotzpi)))
- {
- result_re->val -= M_PI * GSL_REAL(cotzpi);
- result_im->val -= M_PI * GSL_IMAG(cotzpi);
- return ret_val;
- }
- else
- {
- GSL_ERROR("singularity", GSL_EDOM);
- }
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_psi_int(const int n)
- {
- EVAL_RESULT(gsl_sf_psi_int_e(n, &result));
- }
- double gsl_sf_psi(const double x)
- {
- EVAL_RESULT(gsl_sf_psi_e(x, &result));
- }
- double gsl_sf_psi_1piy(const double x)
- {
- EVAL_RESULT(gsl_sf_psi_1piy_e(x, &result));
- }
- double gsl_sf_psi_1_int(const int n)
- {
- EVAL_RESULT(gsl_sf_psi_1_int_e(n, &result));
- }
- double gsl_sf_psi_1(const double x)
- {
- EVAL_RESULT(gsl_sf_psi_1_e(x, &result));
- }
- double gsl_sf_psi_n(const int n, const double x)
- {
- EVAL_RESULT(gsl_sf_psi_n_e(n, x, &result));
- }
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