| /* specfunc/psi.c |
| * |
| * 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 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_exp.h> |
| #include <gsl/gsl_sf_gamma.h> |
| #include <gsl/gsl_sf_zeta.h> |
| #include <gsl/gsl_sf_psi.h> |
| #include <gsl/gsl_complex_math.h> |
| |
| #include <stdio.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, |
| 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)); |
| |
| /* 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); |
| } |
| |
| result_re->val = GSL_REAL(a); |
| result_im->val = GSL_IMAG(a); |
| result_re->err = 2.0 * (1.0 + n_recurse) * GSL_DBL_EPSILON * fabs(result_re->val); |
| result_im->err = 2.0 * (1.0 + n_recurse) * 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 "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)); |
| } |