/* specfunc/legendre_con.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
* Copyright (C) 2010 Brian Gough
*
* 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_poly.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_trig.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_ellint.h>
#include <gsl/gsl_sf_pow_int.h>
#include <gsl/gsl_sf_bessel.h>
#include <gsl/gsl_sf_hyperg.h>
#include <gsl/gsl_sf_legendre.h>
#include "error.h"
#include "legendre.h"
#define Root_2OverPi_ 0.797884560802865355879892
#define locEPS (1000.0*GSL_DBL_EPSILON)
/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
#define RECURSE_LARGE (1.0e-5*GSL_DBL_MAX)
#define RECURSE_SMALL (1.0e+5*GSL_DBL_MIN)
/* Continued fraction for f_{ell+1}/f_ell
* f_ell := P^{-mu-ell}_{-1/2 + I tau}(x), x < 1.0
*
* Uses standard CF method from Temme's book.
*/
static
int
conicalP_negmu_xlt1_CF1(const double mu, const int ell, const double tau,
const double x, gsl_sf_result * result)
{
const double RECUR_BIG = GSL_SQRT_DBL_MAX;
const int maxiter = 5000;
int n = 1;
double xi = x/(sqrt(1.0-x)*sqrt(1.0+x));
double Anm2 = 1.0;
double Bnm2 = 0.0;
double Anm1 = 0.0;
double Bnm1 = 1.0;
double a1 = 1.0;
double b1 = 2.0*(mu + ell + 1.0) * xi;
double An = b1*Anm1 + a1*Anm2;
double Bn = b1*Bnm1 + a1*Bnm2;
double an, bn;
double fn = An/Bn;
while(n < maxiter) {
double old_fn;
double del;
n++;
Anm2 = Anm1;
Bnm2 = Bnm1;
Anm1 = An;
Bnm1 = Bn;
an = tau*tau + (mu - 0.5 + ell + n)*(mu - 0.5 + ell + n);
bn = 2.0*(ell + mu + n) * xi;
An = bn*Anm1 + an*Anm2;
Bn = bn*Bnm1 + an*Bnm2;
if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {
An /= RECUR_BIG;
Bn /= RECUR_BIG;
Anm1 /= RECUR_BIG;
Bnm1 /= RECUR_BIG;
Anm2 /= RECUR_BIG;
Bnm2 /= RECUR_BIG;
}
old_fn = fn;
fn = An/Bn;
del = old_fn/fn;
if(fabs(del - 1.0) < 2.0*GSL_DBL_EPSILON) break;
}
result->val = fn;
result->err = 4.0 * GSL_DBL_EPSILON * (sqrt(n) + 1.0) * fabs(fn);
if(n >= maxiter)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/* Continued fraction for f_{ell+1}/f_ell
* f_ell := P^{-mu-ell}_{-1/2 + I tau}(x), x >= 1.0
*
* Uses Gautschi (Euler) equivalent series.
*/
static
int
conicalP_negmu_xgt1_CF1(const double mu, const int ell, const double tau,
const double x, gsl_sf_result * result)
{
const int maxk = 20000;
const double gamma = 1.0-1.0/(x*x);
const double pre = sqrt(x-1.0)*sqrt(x+1.0) / (x*(2.0*(ell+mu+1.0)));
double tk = 1.0;
double sum = 1.0;
double rhok = 0.0;
int k;
for(k=1; k<maxk; k++) {
double tlk = 2.0*(ell + mu + k);
double l1k = (ell + mu - 0.5 + 1.0 + k);
double ak = -(tau*tau + l1k*l1k)/(tlk*(tlk+2.0)) * gamma;
rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
tk *= rhok;
sum += tk;
if(fabs(tk/sum) < GSL_DBL_EPSILON) break;
}
result->val = pre * sum;
result->err = fabs(pre * tk);
result->err += 2.0 * GSL_DBL_EPSILON * (sqrt(k) + 1.0) * fabs(pre*sum);
if(k >= maxk)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/* Implementation of large negative mu asymptotic
* [Dunster, Proc. Roy. Soc. Edinburgh 119A, 311 (1991), p. 326]
*/
inline
static double olver_U1(double beta2, double p)
{
return (p-1.0)/(24.0*(1.0+beta2)) * (3.0 + beta2*(2.0 + 5.0*p*(1.0+p)));
}
inline
static double olver_U2(double beta2, double p)
{
double beta4 = beta2*beta2;
double p2 = p*p;
double poly1 = 4.0*beta4 + 84.0*beta2 - 63.0;
double poly2 = 16.0*beta4 + 90.0*beta2 - 81.0;
double poly3 = beta2*p2*(97.0*beta2 - 432.0 + 77.0*p*(beta2-6.0) - 385.0*beta2*p2*(1.0 + p));
return (1.0-p)/(1152.0*(1.0+beta2)) * (poly1 + poly2 + poly3);
}
static const double U3c1[] = { -1307.0, -1647.0, 3375.0, 3675.0 };
static const double U3c2[] = { 29366.0, 35835.0, -252360.0, -272630.0,
276810.0, 290499.0 };
static const double U3c3[] = { -29748.0, -8840.0, 1725295.0, 1767025.0,
-7313470.0, -754778.0, 6309875.0, 6480045.0 };
static const double U3c4[] = { 2696.0, -16740.0, -524250.0, -183975.0,
14670540.0, 14172939.0, -48206730.0, -48461985.0,
36756720.0, 37182145.0 };
static const double U3c5[] = { 9136.0, 22480.0, 12760.0,
-252480.0, -4662165.0, -1705341.0,
92370135.0, 86244015.0, -263678415.0,
-260275015.0, 185910725.0, 185910725.0 };
#if 0
static double olver_U3(double beta2, double p)
{
double beta4 = beta2*beta2;
double beta6 = beta4*beta2;
double opb2s = (1.0+beta2)*(1.0+beta2);
double den = 39813120.0 * opb2s*opb2s;
double poly1 = gsl_poly_eval(U3c1, 4, p);
double poly2 = gsl_poly_eval(U3c2, 6, p);
double poly3 = gsl_poly_eval(U3c3, 8, p);
double poly4 = gsl_poly_eval(U3c4, 10, p);
double poly5 = gsl_poly_eval(U3c5, 12, p);
return (p-1.0)*( 1215.0*poly1 + 324.0*beta2*poly2
+ 54.0*beta4*poly3 + 12.0*beta6*poly4
+ beta4*beta4*poly5
) / den;
}
#endif /* 0 */
/* Large negative mu asymptotic
* P^{-mu}_{-1/2 + I tau}, mu -> Inf
* |x| < 1
*
* [Dunster, Proc. Roy. Soc. Edinburgh 119A, 311 (1991), p. 326]
*/
int
gsl_sf_conicalP_xlt1_large_neg_mu_e(double mu, double tau, double x,
gsl_sf_result * result, double * ln_multiplier)
{
double beta = tau/mu;
double beta2 = beta*beta;
double S = beta * acos((1.0-beta2)/(1.0+beta2));
double p = x/sqrt(beta2*(1.0-x*x) + 1.0);
gsl_sf_result lg_mup1;
int lg_stat = gsl_sf_lngamma_e(mu+1.0, &lg_mup1);
double ln_pre_1 = 0.5*mu*(S - log(1.0+beta2) + log((1.0-p)/(1.0+p))) - lg_mup1.val;
double ln_pre_2 = -0.25 * log(1.0 + beta2*(1.0-x));
double ln_pre_3 = -tau * atan(p*beta);
double ln_pre = ln_pre_1 + ln_pre_2 + ln_pre_3;
double sum = 1.0 - olver_U1(beta2, p)/mu + olver_U2(beta2, p)/(mu*mu);
if(sum == 0.0) {
result->val = 0.0;
result->err = 0.0;
*ln_multiplier = 0.0;
return GSL_SUCCESS;
}
else {
int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result);
if(stat_e != GSL_SUCCESS) {
result->val = sum;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum);
*ln_multiplier = ln_pre;
}
else {
*ln_multiplier = 0.0;
}
return lg_stat;
}
}
/* Implementation of large tau asymptotic
*
* A_n^{-mu}, B_n^{-mu} [Olver, p.465, 469]
*/
inline
static double olver_B0_xi(double mu, double xi)
{
return (1.0 - 4.0*mu*mu)/(8.0*xi) * (1.0/tanh(xi) - 1.0/xi);
}
static double olver_A1_xi(double mu, double xi, double x)
{
double B = olver_B0_xi(mu, xi);
double psi;
if(fabs(x - 1.0) < GSL_ROOT4_DBL_EPSILON) {
double y = x - 1.0;
double s = -1.0/3.0 + y*(2.0/15.0 - y *(61.0/945.0 - 452.0/14175.0*y));
psi = (4.0*mu*mu - 1.0)/16.0 * s;
}
else {
psi = (4.0*mu*mu - 1.0)/16.0 * (1.0/(x*x-1.0) - 1.0/(xi*xi));
}
return 0.5*xi*xi*B*B + (mu+0.5)*B - psi + mu/6.0*(0.25 - mu*mu);
}
inline
static double olver_B0_th(double mu, double theta)
{
return -(1.0 - 4.0*mu*mu)/(8.0*theta) * (1.0/tan(theta) - 1.0/theta);
}
static double olver_A1_th(double mu, double theta, double x)
{
double B = olver_B0_th(mu, theta);
double psi;
if(fabs(x - 1.0) < GSL_ROOT4_DBL_EPSILON) {
double y = 1.0 - x;
double s = -1.0/3.0 + y*(2.0/15.0 - y *(61.0/945.0 - 452.0/14175.0*y));
psi = (4.0*mu*mu - 1.0)/16.0 * s;
}
else {
psi = (4.0*mu*mu - 1.0)/16.0 * (1.0/(x*x-1.0) + 1.0/(theta*theta));
}
return -0.5*theta*theta*B*B + (mu+0.5)*B - psi + mu/6.0*(0.25 - mu*mu);
}
/* Large tau uniform asymptotics
* P^{-mu}_{-1/2 + I tau}
* 1 < x
* tau -> Inf
* [Olver, p. 469]
*/
int
gsl_sf_conicalP_xgt1_neg_mu_largetau_e(const double mu, const double tau,
const double x, double acosh_x,
gsl_sf_result * result, double * ln_multiplier)
{
double xi = acosh_x;
double ln_xi_pre;
double ln_pre;
double sumA, sumB, sum;
double arg;
gsl_sf_result J_mup1;
gsl_sf_result J_mu;
double J_mum1;
if(xi < GSL_ROOT4_DBL_EPSILON) {
ln_xi_pre = -xi*xi/6.0; /* log(1.0 - xi*xi/6.0) */
}
else {
gsl_sf_result lnshxi;
gsl_sf_lnsinh_e(xi, &lnshxi);
ln_xi_pre = log(xi) - lnshxi.val; /* log(xi/sinh(xi) */
}
ln_pre = 0.5*ln_xi_pre - mu*log(tau);
arg = tau*xi;
gsl_sf_bessel_Jnu_e(mu + 1.0, arg, &J_mup1);
gsl_sf_bessel_Jnu_e(mu, arg, &J_mu);
J_mum1 = -J_mup1.val + 2.0*mu/arg*J_mu.val; /* careful of mu < 1 */
sumA = 1.0 - olver_A1_xi(-mu, xi, x)/(tau*tau);
sumB = olver_B0_xi(-mu, xi);
sum = J_mu.val * sumA - xi/tau * J_mum1 * sumB;
if(sum == 0.0) {
result->val = 0.0;
result->err = 0.0;
*ln_multiplier = 0.0;
return GSL_SUCCESS;
}
else {
int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result);
if(stat_e != GSL_SUCCESS) {
result->val = sum;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(sum);
*ln_multiplier = ln_pre;
}
else {
*ln_multiplier = 0.0;
}
return GSL_SUCCESS;
}
}
/* Large tau uniform asymptotics
* P^{-mu}_{-1/2 + I tau}
* -1 < x < 1
* tau -> Inf
* [Olver, p. 473]
*/
int
gsl_sf_conicalP_xlt1_neg_mu_largetau_e(const double mu, const double tau,
const double x, const double acos_x,
gsl_sf_result * result, double * ln_multiplier)
{
double theta = acos_x;
double ln_th_pre;
double ln_pre;
double sumA, sumB, sum, sumerr;
double arg;
gsl_sf_result I_mup1, I_mu;
double I_mum1;
if(theta < GSL_ROOT4_DBL_EPSILON) {
ln_th_pre = theta*theta/6.0; /* log(1.0 + theta*theta/6.0) */
}
else {
ln_th_pre = log(theta/sin(theta));
}
ln_pre = 0.5 * ln_th_pre - mu * log(tau);
arg = tau*theta;
gsl_sf_bessel_Inu_e(mu + 1.0, arg, &I_mup1);
gsl_sf_bessel_Inu_e(mu, arg, &I_mu);
I_mum1 = I_mup1.val + 2.0*mu/arg * I_mu.val; /* careful of mu < 1 */
sumA = 1.0 - olver_A1_th(-mu, theta, x)/(tau*tau);
sumB = olver_B0_th(-mu, theta);
sum = I_mu.val * sumA - theta/tau * I_mum1 * sumB;
sumerr = fabs(I_mu.err * sumA);
sumerr += fabs(I_mup1.err * theta/tau * sumB);
sumerr += fabs(I_mu.err * theta/tau * sumB * 2.0 * mu/arg);
if(sum == 0.0) {
result->val = 0.0;
result->err = 0.0;
*ln_multiplier = 0.0;
return GSL_SUCCESS;
}
else {
int stat_e = gsl_sf_exp_mult_e(ln_pre, sum, result);
if(stat_e != GSL_SUCCESS) {
result->val = sum;
result->err = sumerr;
result->err += GSL_DBL_EPSILON * fabs(sum);
*ln_multiplier = ln_pre;
}
else {
*ln_multiplier = 0.0;
}
return GSL_SUCCESS;
}
}
/* Hypergeometric function which appears in the
* large x expansion below:
*
* 2F1(1/4 - mu/2 - I tau/2, 3/4 - mu/2 - I tau/2, 1 - I tau, y)
*
* Note that for the usage below y = 1/x^2;
*/
static
int
conicalP_hyperg_large_x(const double mu, const double tau, const double y,
double * reF, double * imF)
{
const int kmax = 1000;
const double re_a = 0.25 - 0.5*mu;
const double re_b = 0.75 - 0.5*mu;
const double re_c = 1.0;
const double im_a = -0.5*tau;
const double im_b = -0.5*tau;
const double im_c = -tau;
double re_sum = 1.0;
double im_sum = 0.0;
double re_term = 1.0;
double im_term = 0.0;
int k;
for(k=1; k<=kmax; k++) {
double re_ak = re_a + k - 1.0;
double re_bk = re_b + k - 1.0;
double re_ck = re_c + k - 1.0;
double im_ak = im_a;
double im_bk = im_b;
double im_ck = im_c;
double den = re_ck*re_ck + im_ck*im_ck;
double re_multiplier = ((re_ak*re_bk - im_ak*im_bk)*re_ck + im_ck*(im_ak*re_bk + re_ak*im_bk)) / den;
double im_multiplier = ((im_ak*re_bk + re_ak*im_bk)*re_ck - im_ck*(re_ak*re_bk - im_ak*im_bk)) / den;
double re_tmp = re_multiplier*re_term - im_multiplier*im_term;
double im_tmp = im_multiplier*re_term + re_multiplier*im_term;
double asum = fabs(re_sum) + fabs(im_sum);
re_term = y/k * re_tmp;
im_term = y/k * im_tmp;
if(fabs(re_term/asum) < GSL_DBL_EPSILON && fabs(im_term/asum) < GSL_DBL_EPSILON) break;
re_sum += re_term;
im_sum += im_term;
}
*reF = re_sum;
*imF = im_sum;
if(k == kmax)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/* P^{mu}_{-1/2 + I tau}
* x->Inf
*/
int
gsl_sf_conicalP_large_x_e(const double mu, const double tau, const double x,
gsl_sf_result * result, double * ln_multiplier)
{
/* 2F1 term
*/
double y = ( x < 0.5*GSL_SQRT_DBL_MAX ? 1.0/(x*x) : 0.0 );
double reF, imF;
int stat_F = conicalP_hyperg_large_x(mu, tau, y, &reF, &imF);
/* f = Gamma(+i tau)/Gamma(1/2 - mu + i tau)
* FIXME: shift so it's better for tau-> 0
*/
gsl_sf_result lgr_num, lgth_num;
gsl_sf_result lgr_den, lgth_den;
int stat_gn = gsl_sf_lngamma_complex_e(0.0,tau,&lgr_num,&lgth_num);
int stat_gd = gsl_sf_lngamma_complex_e(0.5-mu,tau,&lgr_den,&lgth_den);
double angle = lgth_num.val - lgth_den.val + atan2(imF,reF);
double lnx = log(x);
double lnxp1 = log(x+1.0);
double lnxm1 = log(x-1.0);
double lnpre_const = 0.5*M_LN2 - 0.5*M_LNPI;
double lnpre_comm = (mu-0.5)*lnx - 0.5*mu*(lnxp1 + lnxm1);
double lnpre_err = GSL_DBL_EPSILON * (0.5*M_LN2 + 0.5*M_LNPI)
+ GSL_DBL_EPSILON * fabs((mu-0.5)*lnx)
+ GSL_DBL_EPSILON * fabs(0.5*mu)*(fabs(lnxp1)+fabs(lnxm1));
/* result = pre*|F|*|f| * cos(angle - tau * (log(x)+M_LN2))
*/
gsl_sf_result cos_result;
int stat_cos = gsl_sf_cos_e(angle + tau*(log(x) + M_LN2), &cos_result);
int status = GSL_ERROR_SELECT_4(stat_cos, stat_gd, stat_gn, stat_F);
if(cos_result.val == 0.0) {
result->val = 0.0;
result->err = 0.0;
return status;
}
else {
double lnFf_val = 0.5*log(reF*reF+imF*imF) + lgr_num.val - lgr_den.val;
double lnFf_err = lgr_num.err + lgr_den.err + GSL_DBL_EPSILON * fabs(lnFf_val);
double lnnoc_val = lnpre_const + lnpre_comm + lnFf_val;
double lnnoc_err = lnpre_err + lnFf_err + GSL_DBL_EPSILON * fabs(lnnoc_val);
int stat_e = gsl_sf_exp_mult_err_e(lnnoc_val, lnnoc_err,
cos_result.val, cos_result.err,
result);
if(stat_e == GSL_SUCCESS) {
*ln_multiplier = 0.0;
}
else {
result->val = cos_result.val;
result->err = cos_result.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
*ln_multiplier = lnnoc_val;
}
return status;
}
}
/* P^{mu}_{-1/2 + I tau} first hypergeometric representation
* -1 < x < 1
* This is more effective for |x| small, however it will work w/o
* reservation for any x < 0 because everything is positive
* definite in that case.
*
* [Kolbig, (3)] (note typo in args of gamma functions)
* [Bateman, (22)] (correct form)
*/
static
int
conicalP_xlt1_hyperg_A(double mu, double tau, double x, gsl_sf_result * result)
{
double x2 = x*x;
double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x)));
double pre_val = M_SQRTPI / pow(0.5*sqrt(1-x2), mu);
double pre_err = err_amp * GSL_DBL_EPSILON * (fabs(mu)+1.0) * fabs(pre_val) ;
gsl_sf_result ln_g1, ln_g2, arg_g1, arg_g2;
gsl_sf_result F1, F2;
gsl_sf_result pre1, pre2;
double t1_val, t1_err;
double t2_val, t2_err;
int stat_F1 = gsl_sf_hyperg_2F1_conj_e(0.25 - 0.5*mu, 0.5*tau, 0.5, x2, &F1);
int stat_F2 = gsl_sf_hyperg_2F1_conj_e(0.75 - 0.5*mu, 0.5*tau, 1.5, x2, &F2);
int status = GSL_ERROR_SELECT_2(stat_F1, stat_F2);
gsl_sf_lngamma_complex_e(0.75 - 0.5*mu, -0.5*tau, &ln_g1, &arg_g1);
gsl_sf_lngamma_complex_e(0.25 - 0.5*mu, -0.5*tau, &ln_g2, &arg_g2);
gsl_sf_exp_err_e(-2.0*ln_g1.val, 2.0*ln_g1.err, &pre1);
gsl_sf_exp_err_e(-2.0*ln_g2.val, 2.0*ln_g2.err, &pre2);
pre2.val *= -2.0*x;
pre2.err *= 2.0*fabs(x);
pre2.err += GSL_DBL_EPSILON * fabs(pre2.val);
t1_val = pre1.val * F1.val;
t1_err = fabs(pre1.val) * F1.err + pre1.err * fabs(F1.val);
t2_val = pre2.val * F2.val;
t2_err = fabs(pre2.val) * F2.err + pre2.err * fabs(F2.val);
result->val = pre_val * (t1_val + t2_val);
result->err = pre_val * (t1_err + t2_err);
result->err += pre_err * fabs(t1_val + t2_val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return status;
}
/* P^{mu}_{-1/2 + I tau}
* defining hypergeometric representation
* [Abramowitz+Stegun, 8.1.2]
* 1 < x < 3
* effective for x near 1
*
*/
#if 0
static
int
conicalP_def_hyperg(double mu, double tau, double x, double * result)
{
double F;
int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5, tau, 1.0-mu, 0.5*(1.0-x), &F);
*result = pow((x+1.0)/(x-1.0), 0.5*mu) * F;
return stat_F;
}
#endif /* 0 */
/* P^{mu}_{-1/2 + I tau} second hypergeometric representation
* [Zhurina+Karmazina, (3.1)]
* -1 < x < 3
* effective for x near 1
*
*/
#if 0
static
int
conicalP_xnear1_hyperg_C(double mu, double tau, double x, double * result)
{
double ln_pre, arg_pre;
double ln_g1, arg_g1;
double ln_g2, arg_g2;
double F;
int stat_F = gsl_sf_hyperg_2F1_conj_renorm_e(0.5+mu, tau, 1.0+mu, 0.5*(1.0-x), &F);
gsl_sf_lngamma_complex_e(0.5+mu, tau, &ln_g1, &arg_g1);
gsl_sf_lngamma_complex_e(0.5-mu, tau, &ln_g2, &arg_g2);
ln_pre = mu*M_LN2 - 0.5*mu*log(fabs(x*x-1.0)) + ln_g1 - ln_g2;
arg_pre = arg_g1 - arg_g2;
*result = exp(ln_pre) * F;
return stat_F;
}
#endif /* 0 */
/* V0, V1 from Kolbig, m = 0
*/
static
int
conicalP_0_V(const double t, const double f, const double tau, const double sgn,
double * V0, double * V1)
{
double C[8];
double T[8];
double H[8];
double V[12];
int i;
T[0] = 1.0;
H[0] = 1.0;
V[0] = 1.0;
for(i=1; i<=7; i++) {
T[i] = T[i-1] * t;
H[i] = H[i-1] * (t*f);
}
for(i=1; i<=11; i++) {
V[i] = V[i-1] * tau;
}
C[0] = 1.0;
C[1] = (H[1]-1.0)/(8.0*T[1]);
C[2] = (9.0*H[2] + 6.0*H[1] - 15.0 - sgn*8.0*T[2])/(128.0*T[2]);
C[3] = 5.0*(15.0*H[3] + 27.0*H[2] + 21.0*H[1] - 63.0 - sgn*T[2]*(16.0*H[1]+24.0))/(1024.0*T[3]);
C[4] = 7.0*(525.0*H[4] + 1500.0*H[3] + 2430.0*H[2] + 1980.0*H[1] - 6435.0
+ 192.0*T[4] - sgn*T[2]*(720.0*H[2]+1600.0*H[1]+2160.0)
) / (32768.0*T[4]);
C[5] = 21.0*(2835.0*H[5] + 11025.0*H[4] + 24750.0*H[3] + 38610.0*H[2]
+ 32175.0*H[1] - 109395.0 + T[4]*(1984.0*H[1]+4032.0)
- sgn*T[2]*(4800.0*H[3]+15120.0*H[2]+26400.0*H[1]+34320.0)
) / (262144.0*T[5]);
C[6] = 11.0*(218295.0*H[6] + 1071630.0*H[5] + 3009825.0*H[4] + 6142500.0*H[3]
+ 9398025.0*H[2] + 7936110.0*H[1] - 27776385.0
+ T[4]*(254016.0*H[2]+749952.0*H[1]+1100736.0)
- sgn*T[2]*(441000.0*H[4] + 1814400.0*H[3] + 4127760.0*H[2]
+ 6552000.0*H[1] + 8353800.0 + 31232.0*T[4]
)
) / (4194304.0*T[6]);
*V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4]
+ (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8]
+ sgn * (-C[2]/V[2]
+ (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6]
+ (-1920.0*C[6]/T[4])/V[10]
);
*V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5]
+ (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9]
+ sgn * ((2.0*C[2]/T[1]-C[3])/V[3]
+ (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7]
+ (3840.0*C[6]/T[5])/V[11]
);
return GSL_SUCCESS;
}
/* V0, V1 from Kolbig, m = 1
*/
static
int
conicalP_1_V(const double t, const double f, const double tau, const double sgn,
double * V0, double * V1)
{
double Cm1;
double C[8];
double T[8];
double H[8];
double V[12];
int i;
T[0] = 1.0;
H[0] = 1.0;
V[0] = 1.0;
for(i=1; i<=7; i++) {
T[i] = T[i-1] * t;
H[i] = H[i-1] * (t*f);
}
for(i=1; i<=11; i++) {
V[i] = V[i-1] * tau;
}
Cm1 = -1.0;
C[0] = 3.0*(1.0-H[1])/(8.0*T[1]);
C[1] = (-15.0*H[2]+6.0*H[1]+9.0+sgn*8.0*T[2])/(128.0*T[2]);
C[2] = 3.0*(-35.0*H[3] - 15.0*H[2] + 15.0*H[1] + 35.0 + sgn*T[2]*(32.0*H[1]+8.0))/(1024.0*T[3]);
C[3] = (-4725.0*H[4] - 6300.0*H[3] - 3150.0*H[2] + 3780.0*H[1] + 10395.0
-1216.0*T[4] + sgn*T[2]*(6000.0*H[2]+5760.0*H[1]+1680.0)) / (32768.0*T[4]);
C[4] = 7.0*(-10395.0*H[5] - 23625.0*H[4] - 28350.0*H[3] - 14850.0*H[2]
+19305.0*H[1] + 57915.0 - T[4]*(6336.0*H[1]+6080.0)
+ sgn*T[2]*(16800.0*H[3] + 30000.0*H[2] + 25920.0*H[1] + 7920.0)
) / (262144.0*T[5]);
C[5] = (-2837835.0*H[6] - 9168390.0*H[5] - 16372125.0*H[4] - 18918900*H[3]
-10135125.0*H[2] + 13783770.0*H[1] + 43648605.0
-T[4]*(3044160.0*H[2] + 5588352.0*H[1] + 4213440.0)
+sgn*T[2]*(5556600.0*H[4] + 14817600.0*H[3] + 20790000.0*H[2]
+ 17297280.0*H[1] + 5405400.0 + 323072.0*T[4]
)
) / (4194304.0*T[6]);
C[6] = 0.0;
*V0 = C[0] + (-4.0*C[3]/T[1]+C[4])/V[4]
+ (-192.0*C[5]/T[3]+144.0*C[6]/T[2])/V[8]
+ sgn * (-C[2]/V[2]
+ (-24.0*C[4]/T[2]+12.0*C[5]/T[1]-C[6])/V[6]
);
*V1 = C[1]/V[1] + (8.0*(C[3]/T[2]-C[4]/T[1])+C[5])/V[5]
+ (384.0*C[5]/T[4] - 768.0*C[6]/T[3])/V[9]
+ sgn * (Cm1*V[1] + (2.0*C[2]/T[1]-C[3])/V[3]
+ (48.0*C[4]/T[3]-72.0*C[5]/T[2] + 18.0*C[6]/T[1])/V[7]
);
return GSL_SUCCESS;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
/* P^0_{-1/2 + I lambda}
*/
int
gsl_sf_conicalP_0_e(const double lambda, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= -1.0) {
DOMAIN_ERROR(result);
}
else if(x == 1.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(lambda == 0.0) {
gsl_sf_result K;
int stat_K;
if(x < 1.0) {
const double th = acos(x);
const double s = sin(0.5*th);
stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K);
result->val = 2.0/M_PI * K.val;
result->err = 2.0/M_PI * K.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_K;
}
else {
const double xi = acosh(x);
const double c = cosh(0.5*xi);
const double t = tanh(0.5*xi);
stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K);
result->val = 2.0/M_PI / c * K.val;
result->err = 2.0/M_PI / c * K.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_K;
}
}
else if( (x <= 0.0 && lambda < 1000.0)
|| (x < 0.1 && lambda < 17.0)
|| (x < 0.2 && lambda < 5.0 )
) {
return conicalP_xlt1_hyperg_A(0.0, lambda, x, result);
}
else if( (x <= 0.2 && lambda < 17.0)
|| (x <= 1.5 && lambda < 20.0)
) {
return gsl_sf_hyperg_2F1_conj_e(0.5, lambda, 1.0, (1.0-x)/2, result);
}
else if(1.5 < x && lambda < GSL_MAX(x,20.0)) {
gsl_sf_result P;
double lm;
int stat_P = gsl_sf_conicalP_large_x_e(0.0, lambda, x,
&P, &lm
);
int stat_e = gsl_sf_exp_mult_err_e(lm, 2.0*GSL_DBL_EPSILON * fabs(lm),
P.val, P.err,
result);
return GSL_ERROR_SELECT_2(stat_e, stat_P);
}
else {
double V0, V1;
if(x < 1.0) {
double th = acos(x);
double sth = sqrt(1.0-x*x); /* sin(th) */
gsl_sf_result I0, I1;
int stat_I0 = gsl_sf_bessel_I0_scaled_e(th * lambda, &I0);
int stat_I1 = gsl_sf_bessel_I1_scaled_e(th * lambda, &I1);
int stat_I = GSL_ERROR_SELECT_2(stat_I0, stat_I1);
int stat_V = conicalP_0_V(th, x/sth, lambda, -1.0, &V0, &V1);
double bessterm = V0 * I0.val + V1 * I1.val;
double besserr = fabs(V0) * I0.err + fabs(V1) * I1.err;
double arg1 = th*lambda;
double sqts = sqrt(th/sth);
int stat_e = gsl_sf_exp_mult_err_e(arg1, 4.0 * GSL_DBL_EPSILON * fabs(arg1),
sqts * bessterm, sqts * besserr,
result);
return GSL_ERROR_SELECT_3(stat_e, stat_V, stat_I);
}
else {
double sh = sqrt(x-1.0)*sqrt(x+1.0); /* sinh(xi) */
double xi = log(x + sh); /* xi = acosh(x) */
gsl_sf_result J0, J1;
int stat_J0 = gsl_sf_bessel_J0_e(xi * lambda, &J0);
int stat_J1 = gsl_sf_bessel_J1_e(xi * lambda, &J1);
int stat_J = GSL_ERROR_SELECT_2(stat_J0, stat_J1);
int stat_V = conicalP_0_V(xi, x/sh, lambda, 1.0, &V0, &V1);
double bessterm = V0 * J0.val + V1 * J1.val;
double besserr = fabs(V0) * J0.err + fabs(V1) * J1.err;
double pre_val = sqrt(xi/sh);
double pre_err = 2.0 * fabs(pre_val);
result->val = pre_val * bessterm;
result->err = pre_val * besserr;
result->err += pre_err * fabs(bessterm);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_V, stat_J);
}
}
}
/* P^1_{-1/2 + I lambda}
*/
int
gsl_sf_conicalP_1_e(const double lambda, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= -1.0) {
DOMAIN_ERROR(result);
}
else if(lambda == 0.0) {
gsl_sf_result K, E;
int stat_K, stat_E;
if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x < 1.0) {
if(1.0-x < GSL_SQRT_DBL_EPSILON) {
double err_amp = GSL_MAX_DBL(1.0, 1.0/(GSL_DBL_EPSILON + fabs(1.0-x)));
result->val = 0.25/M_SQRT2 * sqrt(1.0-x) * (1.0 + 5.0/16.0 * (1.0-x));
result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
const double th = acos(x);
const double s = sin(0.5*th);
const double c2 = 1.0 - s*s;
const double sth = sin(th);
const double pre = 2.0/(M_PI*sth);
stat_K = gsl_sf_ellint_Kcomp_e(s, GSL_MODE_DEFAULT, &K);
stat_E = gsl_sf_ellint_Ecomp_e(s, GSL_MODE_DEFAULT, &E);
result->val = pre * (E.val - c2 * K.val);
result->err = pre * (E.err + fabs(c2) * K.err);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_K, stat_E);
}
}
else {
if(x-1.0 < GSL_SQRT_DBL_EPSILON) {
double err_amp = GSL_MAX_DBL(1.0, 1.0/(GSL_DBL_EPSILON + fabs(1.0-x)));
result->val = -0.25/M_SQRT2 * sqrt(x-1.0) * (1.0 - 5.0/16.0 * (x-1.0));
result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
const double xi = acosh(x);
const double c = cosh(0.5*xi);
const double t = tanh(0.5*xi);
const double sxi = sinh(xi);
const double pre = 2.0/(M_PI*sxi) * c;
stat_K = gsl_sf_ellint_Kcomp_e(t, GSL_MODE_DEFAULT, &K);
stat_E = gsl_sf_ellint_Ecomp_e(t, GSL_MODE_DEFAULT, &E);
result->val = pre * (E.val - K.val);
result->err = pre * (E.err + K.err);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_K, stat_E);
}
}
}
else if( (x <= 0.0 && lambda < 1000.0)
|| (x < 0.1 && lambda < 17.0)
|| (x < 0.2 && lambda < 5.0 )
) {
return conicalP_xlt1_hyperg_A(1.0, lambda, x, result);
}
else if( (x <= 0.2 && lambda < 17.0)
|| (x < 1.5 && lambda < 20.0)
) {
const double arg = fabs(x*x - 1.0);
const double sgn = GSL_SIGN(1.0 - x);
const double pre = 0.5*(lambda*lambda + 0.25) * sgn * sqrt(arg);
gsl_sf_result F;
int stat_F = gsl_sf_hyperg_2F1_conj_e(1.5, lambda, 2.0, (1.0-x)/2, &F);
result->val = pre * F.val;
result->err = fabs(pre) * F.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_F;
}
else if(1.5 <= x && lambda < GSL_MAX(x,20.0)) {
gsl_sf_result P;
double lm;
int stat_P = gsl_sf_conicalP_large_x_e(1.0, lambda, x,
&P, &lm
);
int stat_e = gsl_sf_exp_mult_err_e(lm, 2.0 * GSL_DBL_EPSILON * fabs(lm),
P.val, P.err,
result);
return GSL_ERROR_SELECT_2(stat_e, stat_P);
}
else {
double V0, V1;
if(x < 1.0) {
const double sqrt_1mx = sqrt(1.0 - x);
const double sqrt_1px = sqrt(1.0 + x);
const double th = acos(x);
const double sth = sqrt_1mx * sqrt_1px; /* sin(th) */
gsl_sf_result I0, I1;
int stat_I0 = gsl_sf_bessel_I0_scaled_e(th * lambda, &I0);
int stat_I1 = gsl_sf_bessel_I1_scaled_e(th * lambda, &I1);
int stat_I = GSL_ERROR_SELECT_2(stat_I0, stat_I1);
int stat_V = conicalP_1_V(th, x/sth, lambda, -1.0, &V0, &V1);
double bessterm = V0 * I0.val + V1 * I1.val;
double besserr = fabs(V0) * I0.err + fabs(V1) * I1.err
+ 2.0 * GSL_DBL_EPSILON * fabs(V0 * I0.val)
+ 2.0 * GSL_DBL_EPSILON * fabs(V1 * I1.val);
double arg1 = th * lambda;
double sqts = sqrt(th/sth);
int stat_e = gsl_sf_exp_mult_err_e(arg1, 2.0 * GSL_DBL_EPSILON * fabs(arg1),
sqts * bessterm, sqts * besserr,
result);
result->err *= 1.0/sqrt_1mx;
return GSL_ERROR_SELECT_3(stat_e, stat_V, stat_I);
}
else {
const double sqrt_xm1 = sqrt(x - 1.0);
const double sqrt_xp1 = sqrt(x + 1.0);
const double sh = sqrt_xm1 * sqrt_xp1; /* sinh(xi) */
const double xi = log(x + sh); /* xi = acosh(x) */
const double xi_lam = xi * lambda;
gsl_sf_result J0, J1;
const int stat_J0 = gsl_sf_bessel_J0_e(xi_lam, &J0);
const int stat_J1 = gsl_sf_bessel_J1_e(xi_lam, &J1);
const int stat_J = GSL_ERROR_SELECT_2(stat_J0, stat_J1);
const int stat_V = conicalP_1_V(xi, x/sh, lambda, 1.0, &V0, &V1);
const double bessterm = V0 * J0.val + V1 * J1.val;
const double besserr = fabs(V0) * J0.err + fabs(V1) * J1.err
+ 512.0 * 2.0 * GSL_DBL_EPSILON * fabs(V0 * J0.val)
+ 512.0 * 2.0 * GSL_DBL_EPSILON * fabs(V1 * J1.val)
+ GSL_DBL_EPSILON * fabs(xi_lam * V0 * J1.val)
+ GSL_DBL_EPSILON * fabs(xi_lam * V1 * J0.val);
const double pre = sqrt(xi/sh);
result->val = pre * bessterm;
result->err = pre * besserr * sqrt_xp1 / sqrt_xm1;
result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_V, stat_J);
}
}
}
/* P^{1/2}_{-1/2 + I lambda} (x)
* [Abramowitz+Stegun 8.6.8, 8.6.12]
* checked OK [GJ] Fri May 8 12:24:36 MDT 1998
*/
int gsl_sf_conicalP_half_e(const double lambda, const double x,
gsl_sf_result * result
)
{
/* CHECK_POINTER(result) */
if(x <= -1.0) {
DOMAIN_ERROR(result);
}
else if(x < 1.0) {
double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x)));
double ac = acos(x);
double den = sqrt(sqrt(1.0-x)*sqrt(1.0+x));
result->val = Root_2OverPi_ / den * cosh(ac * lambda);
result->err = err_amp * 3.0 * GSL_DBL_EPSILON * fabs(result->val);
result->err *= fabs(ac * lambda) + 1.0;
return GSL_SUCCESS;
}
else if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* x > 1 */
double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x)));
double sq_term = sqrt(x-1.0)*sqrt(x+1.0);
double ln_term = log(x + sq_term);
double den = sqrt(sq_term);
double carg_val = lambda * ln_term;
double carg_err = 2.0 * GSL_DBL_EPSILON * fabs(carg_val);
gsl_sf_result cos_result;
int stat_cos = gsl_sf_cos_err_e(carg_val, carg_err, &cos_result);
result->val = Root_2OverPi_ / den * cos_result.val;
result->err = err_amp * Root_2OverPi_ / den * cos_result.err;
result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_cos;
}
}
/* P^{-1/2}_{-1/2 + I lambda} (x)
* [Abramowitz+Stegun 8.6.9, 8.6.14]
* checked OK [GJ] Fri May 8 12:24:43 MDT 1998
*/
int gsl_sf_conicalP_mhalf_e(const double lambda, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= -1.0) {
DOMAIN_ERROR(result);
}
else if(x < 1.0) {
double ac = acos(x);
double den = sqrt(sqrt(1.0-x)*sqrt(1.0+x));
double arg = ac * lambda;
double err_amp = 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-fabs(x)));
if(fabs(arg) < GSL_SQRT_DBL_EPSILON) {
result->val = Root_2OverPi_ / den * ac;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
result->err *= err_amp;
}
else {
result->val = Root_2OverPi_ / (den*lambda) * sinh(arg);
result->err = GSL_DBL_EPSILON * (fabs(arg)+1.0) * fabs(result->val);
result->err *= err_amp;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
return GSL_SUCCESS;
}
else if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* x > 1 */
double sq_term = sqrt(x-1.0)*sqrt(x+1.0);
double ln_term = log(x + sq_term);
double den = sqrt(sq_term);
double arg_val = lambda * ln_term;
double arg_err = 2.0 * GSL_DBL_EPSILON * fabs(arg_val);
if(arg_val < GSL_SQRT_DBL_EPSILON) {
result->val = Root_2OverPi_ / den * ln_term;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
gsl_sf_result sin_result;
int stat_sin = gsl_sf_sin_err_e(arg_val, arg_err, &sin_result);
result->val = Root_2OverPi_ / (den*lambda) * sin_result.val;
result->err = Root_2OverPi_ / fabs(den*lambda) * sin_result.err;
result->err += 3.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_sin;
}
}
}
int gsl_sf_conicalP_sph_reg_e(const int l, const double lambda,
const double x,
gsl_sf_result * result
)
{
/* CHECK_POINTER(result) */
if(x <= -1.0 || l < -1) {
DOMAIN_ERROR(result);
}
else if(l == -1) {
return gsl_sf_conicalP_half_e(lambda, x, result);
}
else if(l == 0) {
return gsl_sf_conicalP_mhalf_e(lambda, x, result);
}
else if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x < 0.0) {
double c = 1.0/sqrt(1.0-x*x);
gsl_sf_result r_Pellm1;
gsl_sf_result r_Pell;
int stat_0 = gsl_sf_conicalP_half_e(lambda, x, &r_Pellm1); /* P^( 1/2) */
int stat_1 = gsl_sf_conicalP_mhalf_e(lambda, x, &r_Pell); /* P^(-1/2) */
int stat_P = GSL_ERROR_SELECT_2(stat_0, stat_1);
double Pellm1 = r_Pellm1.val;
double Pell = r_Pell.val;
double Pellp1;
int ell;
for(ell=0; ell<l; ell++) {
double d = (ell+1.0)*(ell+1.0) + lambda*lambda;
Pellp1 = (Pellm1 - (2.0*ell+1.0)*c*x * Pell) / d;
Pellm1 = Pell;
Pell = Pellp1;
}
result->val = Pell;
result->err = (0.5*l + 1.0) * GSL_DBL_EPSILON * fabs(Pell);
result->err += GSL_DBL_EPSILON * l * fabs(result->val);
return stat_P;
}
else if(x < 1.0) {
const double xi = x/(sqrt(1.0-x)*sqrt(1.0+x));
gsl_sf_result rat;
gsl_sf_result Phf;
int stat_CF1 = conicalP_negmu_xlt1_CF1(0.5, l, lambda, x, &rat);
int stat_Phf = gsl_sf_conicalP_half_e(lambda, x, &Phf);
double Pellp1 = rat.val * GSL_SQRT_DBL_MIN;
double Pell = GSL_SQRT_DBL_MIN;
double Pellm1;
int ell;
for(ell=l; ell>=0; ell--) {
double d = (ell+1.0)*(ell+1.0) + lambda*lambda;
Pellm1 = (2.0*ell+1.0)*xi * Pell + d * Pellp1;
Pellp1 = Pell;
Pell = Pellm1;
}
result->val = GSL_SQRT_DBL_MIN * Phf.val / Pell;
result->err = GSL_SQRT_DBL_MIN * Phf.err / fabs(Pell);
result->err += fabs(rat.err/rat.val) * (l + 1.0) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_Phf, stat_CF1);
}
else if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* x > 1.0 */
const double xi = x/sqrt((x-1.0)*(x+1.0));
gsl_sf_result rat;
int stat_CF1 = conicalP_negmu_xgt1_CF1(0.5, l, lambda, x, &rat);
int stat_P;
double Pellp1 = rat.val * GSL_SQRT_DBL_MIN;
double Pell = GSL_SQRT_DBL_MIN;
double Pellm1;
int ell;
for(ell=l; ell>=0; ell--) {
double d = (ell+1.0)*(ell+1.0) + lambda*lambda;
Pellm1 = (2.0*ell+1.0)*xi * Pell - d * Pellp1;
Pellp1 = Pell;
Pell = Pellm1;
}
if(fabs(Pell) > fabs(Pellp1)){
gsl_sf_result Phf;
stat_P = gsl_sf_conicalP_half_e(lambda, x, &Phf);
result->val = GSL_SQRT_DBL_MIN * Phf.val / Pell;
result->err = 2.0 * GSL_SQRT_DBL_MIN * Phf.err / fabs(Pell);
result->err += 2.0 * fabs(rat.err/rat.val) * (l + 1.0) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
else {
gsl_sf_result Pmhf;
stat_P = gsl_sf_conicalP_mhalf_e(lambda, x, &Pmhf);
result->val = GSL_SQRT_DBL_MIN * Pmhf.val / Pellp1;
result->err = 2.0 * GSL_SQRT_DBL_MIN * Pmhf.err / fabs(Pellp1);
result->err += 2.0 * fabs(rat.err/rat.val) * (l + 1.0) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
return GSL_ERROR_SELECT_2(stat_P, stat_CF1);
}
}
int gsl_sf_conicalP_cyl_reg_e(const int m, const double lambda,
const double x,
gsl_sf_result * result
)
{
/* CHECK_POINTER(result) */
if(x <= -1.0 || m < -1) {
DOMAIN_ERROR(result);
}
else if(m == -1) {
return gsl_sf_conicalP_1_e(lambda, x, result);
}
else if(m == 0) {
return gsl_sf_conicalP_0_e(lambda, x, result);
}
else if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x < 0.0) {
double c = 1.0/sqrt(1.0-x*x);
gsl_sf_result r_Pkm1;
gsl_sf_result r_Pk;
int stat_0 = gsl_sf_conicalP_1_e(lambda, x, &r_Pkm1); /* P^1 */
int stat_1 = gsl_sf_conicalP_0_e(lambda, x, &r_Pk); /* P^0 */
int stat_P = GSL_ERROR_SELECT_2(stat_0, stat_1);
double Pkm1 = r_Pkm1.val;
double Pk = r_Pk.val;
double Pkp1;
int k;
for(k=0; k<m; k++) {
double d = (k+0.5)*(k+0.5) + lambda*lambda;
Pkp1 = (Pkm1 - 2.0*k*c*x * Pk) / d;
Pkm1 = Pk;
Pk = Pkp1;
}
result->val = Pk;
result->err = (m + 2.0) * GSL_DBL_EPSILON * fabs(Pk);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_P;
}
else if(x < 1.0) {
const double xi = x/(sqrt(1.0-x)*sqrt(1.0+x));
gsl_sf_result rat;
gsl_sf_result P0;
int stat_CF1 = conicalP_negmu_xlt1_CF1(0.0, m, lambda, x, &rat);
int stat_P0 = gsl_sf_conicalP_0_e(lambda, x, &P0);
double Pkp1 = rat.val * GSL_SQRT_DBL_MIN;
double Pk = GSL_SQRT_DBL_MIN;
double Pkm1;
int k;
for(k=m; k>0; k--) {
double d = (k+0.5)*(k+0.5) + lambda*lambda;
Pkm1 = 2.0*k*xi * Pk + d * Pkp1;
Pkp1 = Pk;
Pk = Pkm1;
}
result->val = GSL_SQRT_DBL_MIN * P0.val / Pk;
result->err = 2.0 * GSL_SQRT_DBL_MIN * P0.err / fabs(Pk);
result->err += 2.0 * fabs(rat.err/rat.val) * (m + 1.0) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_P0, stat_CF1);
}
else if(x == 1.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* x > 1.0 */
const double xi = x/sqrt((x-1.0)*(x+1.0));
gsl_sf_result rat;
int stat_CF1 = conicalP_negmu_xgt1_CF1(0.0, m, lambda, x, &rat);
int stat_P;
double Pkp1 = rat.val * GSL_SQRT_DBL_MIN;
double Pk = GSL_SQRT_DBL_MIN;
double Pkm1;
int k;
for(k=m; k>-1; k--) {
double d = (k+0.5)*(k+0.5) + lambda*lambda;
Pkm1 = 2.0*k*xi * Pk - d * Pkp1;
Pkp1 = Pk;
Pk = Pkm1;
}
if(fabs(Pk) > fabs(Pkp1)){
gsl_sf_result P1;
stat_P = gsl_sf_conicalP_1_e(lambda, x, &P1);
result->val = GSL_SQRT_DBL_MIN * P1.val / Pk;
result->err = 2.0 * GSL_SQRT_DBL_MIN * P1.err / fabs(Pk);
result->err += 2.0 * fabs(rat.err/rat.val) * (m+2.0) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
else {
gsl_sf_result P0;
stat_P = gsl_sf_conicalP_0_e(lambda, x, &P0);
result->val = GSL_SQRT_DBL_MIN * P0.val / Pkp1;
result->err = 2.0 * GSL_SQRT_DBL_MIN * P0.err / fabs(Pkp1);
result->err += 2.0 * fabs(rat.err/rat.val) * (m+2.0) * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
return GSL_ERROR_SELECT_2(stat_P, stat_CF1);
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_conicalP_0(const double lambda, const double x)
{
EVAL_RESULT(gsl_sf_conicalP_0_e(lambda, x, &result));
}
double gsl_sf_conicalP_1(const double lambda, const double x)
{
EVAL_RESULT(gsl_sf_conicalP_1_e(lambda, x, &result));
}
double gsl_sf_conicalP_half(const double lambda, const double x)
{
EVAL_RESULT(gsl_sf_conicalP_half_e(lambda, x, &result));
}
double gsl_sf_conicalP_mhalf(const double lambda, const double x)
{
EVAL_RESULT(gsl_sf_conicalP_mhalf_e(lambda, x, &result));
}
double gsl_sf_conicalP_sph_reg(const int l, const double lambda, const double x)
{
EVAL_RESULT(gsl_sf_conicalP_sph_reg_e(l, lambda, x, &result));
}
double gsl_sf_conicalP_cyl_reg(const int m, const double lambda, const double x)
{
EVAL_RESULT(gsl_sf_conicalP_cyl_reg_e(m, lambda, x, &result));
}