/* 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 <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));
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 "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));
}