/* specfunc/legendre_H3d.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 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_trig.h>
#include <gsl/gsl_sf_legendre.h>
#include "error.h"
#include "legendre.h"
/* See [Abbott+Schaefer, Ap.J. 308, 546 (1986)] for
* enough details to follow what is happening here.
*/
/* Logarithm of normalization factor, Log[N(ell,lambda)].
* N(ell,lambda) = Product[ lambda^2 + n^2, {n,0,ell} ]
* = |Gamma(ell + 1 + I lambda)|^2 lambda sinh(Pi lambda) / Pi
* Assumes ell >= 0.
*/
static
int
legendre_H3d_lnnorm(const int ell, const double lambda, double * result)
{
double abs_lam = fabs(lambda);
if(abs_lam == 0.0) {
*result = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else if(lambda > (ell + 1.0)/GSL_ROOT3_DBL_EPSILON) {
/* There is a cancellation between the sinh(Pi lambda)
* term and the log(gamma(ell + 1 + i lambda) in the
* result below, so we show some care and save some digits.
* Note that the above guarantees that lambda is large,
* since ell >= 0. We use Stirling and a simple expansion
* of sinh.
*/
double rat = (ell+1.0)/lambda;
double ln_lam2ell2 = 2.0*log(lambda) + log(1.0 + rat*rat);
double lg_corrected = -2.0*(ell+1.0) + M_LNPI + (ell+0.5)*ln_lam2ell2 + 1.0/(288.0*lambda*lambda);
double angle_terms = lambda * 2.0 * rat * (1.0 - rat*rat/3.0);
*result = log(abs_lam) + lg_corrected + angle_terms - M_LNPI;
return GSL_SUCCESS;
}
else {
gsl_sf_result lg_r;
gsl_sf_result lg_theta;
gsl_sf_result ln_sinh;
gsl_sf_lngamma_complex_e(ell+1.0, lambda, &lg_r, &lg_theta);
gsl_sf_lnsinh_e(M_PI * abs_lam, &ln_sinh);
*result = log(abs_lam) + ln_sinh.val + 2.0*lg_r.val - M_LNPI;
return GSL_SUCCESS;
}
}
/* Calculate series for small eta*lambda.
* Assumes eta > 0, lambda != 0.
*
* This is just the defining hypergeometric for the Legendre function.
*
* P^{mu}_{-1/2 + I lam}(z) = 1/Gamma(l+3/2) ((z+1)/(z-1)^(mu/2)
* 2F1(1/2 - I lam, 1/2 + I lam; l+3/2; (1-z)/2)
* We use
* z = cosh(eta)
* (z-1)/2 = sinh^2(eta/2)
*
* And recall
* H3d = sqrt(Pi Norm /(2 lam^2 sinh(eta))) P^{-l-1/2}_{-1/2 + I lam}(cosh(eta))
*/
static
int
legendre_H3d_series(const int ell, const double lambda, const double eta,
gsl_sf_result * result)
{
const int nmax = 5000;
const double shheta = sinh(0.5*eta);
const double ln_zp1 = M_LN2 + log(1.0 + shheta*shheta);
const double ln_zm1 = M_LN2 + 2.0*log(shheta);
const double zeta = -shheta*shheta;
gsl_sf_result lg_lp32;
double term = 1.0;
double sum = 1.0;
double sum_err = 0.0;
gsl_sf_result lnsheta;
double lnN;
double lnpre_val, lnpre_err, lnprepow;
int stat_e;
int n;
gsl_sf_lngamma_e(ell + 3.0/2.0, &lg_lp32);
gsl_sf_lnsinh_e(eta, &lnsheta);
legendre_H3d_lnnorm(ell, lambda, &lnN);
lnprepow = 0.5*(ell + 0.5) * (ln_zm1 - ln_zp1);
lnpre_val = lnprepow + 0.5*(lnN + M_LNPI - M_LN2 - lnsheta.val) - lg_lp32.val - log(fabs(lambda));
lnpre_err = lnsheta.err + lg_lp32.err + GSL_DBL_EPSILON * fabs(lnpre_val);
lnpre_err += 2.0*GSL_DBL_EPSILON * (fabs(lnN) + M_LNPI + M_LN2);
lnpre_err += 2.0*GSL_DBL_EPSILON * (0.5*(ell + 0.5) * (fabs(ln_zm1) + fabs(ln_zp1)));
for(n=1; n<nmax; n++) {
double aR = n - 0.5;
term *= (aR*aR + lambda*lambda)*zeta/(ell + n + 0.5)/n;
sum += term;
sum_err += 2.0*GSL_DBL_EPSILON*fabs(term);
if(fabs(term/sum) < 2.0 * GSL_DBL_EPSILON) break;
}
stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err, sum, fabs(term)+sum_err, result);
return GSL_ERROR_SELECT_2(stat_e, (n==nmax ? GSL_EMAXITER : GSL_SUCCESS));
}
/* Evaluate legendre_H3d(ell+1)/legendre_H3d(ell)
* by continued fraction.
*/
#if 0
static
int
legendre_H3d_CF1(const int ell, const double lambda, const double coth_eta,
gsl_sf_result * result)
{
const double RECUR_BIG = GSL_SQRT_DBL_MAX;
const int maxiter = 5000;
int n = 1;
double Anm2 = 1.0;
double Bnm2 = 0.0;
double Anm1 = 0.0;
double Bnm1 = 1.0;
double a1 = hypot(lambda, ell+1.0);
double b1 = (2.0*ell + 3.0) * coth_eta;
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 = -(lambda*lambda + ((double)ell + n)*((double)ell + n));
bn = (2.0*ell + 2.0*n + 1.0) * coth_eta;
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) < 4.0*GSL_DBL_EPSILON) break;
}
result->val = fn;
result->err = 2.0 * GSL_DBL_EPSILON * (sqrt(n)+1.0) * fabs(fn);
if(n >= maxiter)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
#endif /* 0 */
/* Evaluate legendre_H3d(ell+1)/legendre_H3d(ell)
* by continued fraction. Use the Gautschi (Euler)
* equivalent series.
*/
/* FIXME: Maybe we have to worry about this. The a_k are
* not positive and there can be a blow-up. It happened
* for J_nu once or twice. Then we should probably use
* the method above.
*/
static
int
legendre_H3d_CF1_ser(const int ell, const double lambda, const double coth_eta,
gsl_sf_result * result)
{
const double pre = hypot(lambda, ell+1.0)/((2.0*ell+3)*coth_eta);
const int maxk = 20000;
double tk = 1.0;
double sum = 1.0;
double rhok = 0.0;
double sum_err = 0.0;
int k;
for(k=1; k<maxk; k++) {
double tlk = (2.0*ell + 1.0 + 2.0*k);
double l1k = (ell + 1.0 + k);
double ak = -(lambda*lambda + l1k*l1k)/(tlk*(tlk+2.0)*coth_eta*coth_eta);
rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
tk *= rhok;
sum += tk;
sum_err += 2.0 * GSL_DBL_EPSILON * k * fabs(tk);
if(fabs(tk/sum) < GSL_DBL_EPSILON) break;
}
result->val = pre * sum;
result->err = fabs(pre * tk);
result->err += fabs(pre * sum_err);
result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val);
if(k >= maxk)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_legendre_H3d_0_e(const double lambda, const double eta, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(eta < 0.0) {
DOMAIN_ERROR(result);
}
else if(eta == 0.0 || lambda == 0.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
const double lam_eta = lambda * eta;
gsl_sf_result s;
gsl_sf_sin_err_e(lam_eta, 2.0*GSL_DBL_EPSILON * fabs(lam_eta), &s);
if(eta > -0.5*GSL_LOG_DBL_EPSILON) {
double f = 2.0 / lambda * exp(-eta);
result->val = f * s.val;
result->err = fabs(f * s.val) * (fabs(eta) + 1.0) * GSL_DBL_EPSILON;
result->err += fabs(f) * s.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
else {
double f = 1.0/(lambda*sinh(eta));
result->val = f * s.val;
result->err = fabs(f * s.val) * (fabs(eta) + 1.0) * GSL_DBL_EPSILON;
result->err += fabs(f) * s.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
}
return GSL_SUCCESS;
}
}
int
gsl_sf_legendre_H3d_1_e(const double lambda, const double eta, gsl_sf_result * result)
{
const double xi = fabs(eta*lambda);
const double lsq = lambda*lambda;
const double lsqp1 = lsq + 1.0;
/* CHECK_POINTER(result) */
if(eta < 0.0) {
DOMAIN_ERROR(result);
}
else if(eta == 0.0 || lambda == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(xi < GSL_ROOT5_DBL_EPSILON && eta < GSL_ROOT5_DBL_EPSILON) {
double etasq = eta*eta;
double xisq = xi*xi;
double term1 = (etasq + xisq)/3.0;
double term2 = -(2.0*etasq*etasq + 5.0*etasq*xisq + 3.0*xisq*xisq)/90.0;
double sinh_term = 1.0 - eta*eta/6.0 * (1.0 - 7.0/60.0*eta*eta);
double pre = sinh_term/sqrt(lsqp1) / eta;
result->val = pre * (term1 + term2);
result->err = pre * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
double sin_term; /* Sin(xi)/xi */
double cos_term; /* Cos(xi) */
double coth_term; /* eta/Tanh(eta) */
double sinh_term; /* eta/Sinh(eta) */
double sin_term_err;
double cos_term_err;
double t1;
double pre_val;
double pre_err;
double term1;
double term2;
if(xi < GSL_ROOT5_DBL_EPSILON) {
sin_term = 1.0 - xi*xi/6.0 * (1.0 - xi*xi/20.0);
cos_term = 1.0 - 0.5*xi*xi * (1.0 - xi*xi/12.0);
sin_term_err = GSL_DBL_EPSILON;
cos_term_err = GSL_DBL_EPSILON;
}
else {
gsl_sf_result sin_xi_result;
gsl_sf_result cos_xi_result;
gsl_sf_sin_e(xi, &sin_xi_result);
gsl_sf_cos_e(xi, &cos_xi_result);
sin_term = sin_xi_result.val/xi;
cos_term = cos_xi_result.val;
sin_term_err = sin_xi_result.err/fabs(xi);
cos_term_err = cos_xi_result.err;
}
if(eta < GSL_ROOT5_DBL_EPSILON) {
coth_term = 1.0 + eta*eta/3.0 * (1.0 - eta*eta/15.0);
sinh_term = 1.0 - eta*eta/6.0 * (1.0 - 7.0/60.0*eta*eta);
}
else {
coth_term = eta/tanh(eta);
sinh_term = eta/sinh(eta);
}
t1 = sqrt(lsqp1) * eta;
pre_val = sinh_term/t1;
pre_err = 2.0 * GSL_DBL_EPSILON * fabs(pre_val);
term1 = sin_term*coth_term;
term2 = cos_term;
result->val = pre_val * (term1 - term2);
result->err = pre_err * fabs(term1 - term2);
result->err += pre_val * (sin_term_err * coth_term + cos_term_err);
result->err += pre_val * fabs(term1-term2) * (fabs(eta) + 1.0) * GSL_DBL_EPSILON;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
int
gsl_sf_legendre_H3d_e(const int ell, const double lambda, const double eta,
gsl_sf_result * result)
{
const double abs_lam = fabs(lambda);
const double lsq = abs_lam*abs_lam;
const double xi = abs_lam * eta;
const double cosh_eta = cosh(eta);
/* CHECK_POINTER(result) */
if(eta < 0.0) {
DOMAIN_ERROR(result);
}
else if(eta > GSL_LOG_DBL_MAX) {
/* cosh(eta) is too big. */
OVERFLOW_ERROR(result);
}
else if(ell == 0) {
return gsl_sf_legendre_H3d_0_e(lambda, eta, result);
}
else if(ell == 1) {
return gsl_sf_legendre_H3d_1_e(lambda, eta, result);
}
else if(eta == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(xi < 1.0) {
return legendre_H3d_series(ell, lambda, eta, result);
}
else if((ell*ell+lsq)/sqrt(1.0+lsq)/(cosh_eta*cosh_eta) < 5.0*GSL_ROOT3_DBL_EPSILON) {
/* Large argument.
*/
gsl_sf_result P;
double lm;
int stat_P = gsl_sf_conicalP_large_x_e(-ell-0.5, lambda, cosh_eta, &P, &lm);
if(P.val == 0.0) {
result->val = 0.0;
result->err = 0.0;
return stat_P;
}
else {
double lnN;
gsl_sf_result lnsh;
double ln_abslam;
double lnpre_val, lnpre_err;
int stat_e;
gsl_sf_lnsinh_e(eta, &lnsh);
legendre_H3d_lnnorm(ell, lambda, &lnN);
ln_abslam = log(abs_lam);
lnpre_val = 0.5*(M_LNPI + lnN - M_LN2 - lnsh.val) - ln_abslam;
lnpre_err = lnsh.err;
lnpre_err += 2.0 * GSL_DBL_EPSILON * (0.5*(M_LNPI + M_LN2 + fabs(lnN)) + fabs(ln_abslam));
lnpre_err += 2.0 * GSL_DBL_EPSILON * fabs(lnpre_val);
stat_e = gsl_sf_exp_mult_err_e(lnpre_val + lm, lnpre_err, P.val, P.err, result);
return GSL_ERROR_SELECT_2(stat_e, stat_P);
}
}
else if(abs_lam > 1000.0*ell*ell) {
/* Large degree.
*/
gsl_sf_result P;
double lm;
int stat_P = gsl_sf_conicalP_xgt1_neg_mu_largetau_e(ell+0.5,
lambda,
cosh_eta, eta,
&P, &lm);
if(P.val == 0.0) {
result->val = 0.0;
result->err = 0.0;
return stat_P;
}
else {
double lnN;
gsl_sf_result lnsh;
double ln_abslam;
double lnpre_val, lnpre_err;
int stat_e;
gsl_sf_lnsinh_e(eta, &lnsh);
legendre_H3d_lnnorm(ell, lambda, &lnN);
ln_abslam = log(abs_lam);
lnpre_val = 0.5*(M_LNPI + lnN - M_LN2 - lnsh.val) - ln_abslam;
lnpre_err = lnsh.err;
lnpre_err += GSL_DBL_EPSILON * (0.5*(M_LNPI + M_LN2 + fabs(lnN)) + fabs(ln_abslam));
lnpre_err += 2.0 * GSL_DBL_EPSILON * fabs(lnpre_val);
stat_e = gsl_sf_exp_mult_err_e(lnpre_val + lm, lnpre_err, P.val, P.err, result);
return GSL_ERROR_SELECT_2(stat_e, stat_P);
}
}
else {
/* Backward recurrence.
*/
const double coth_eta = 1.0/tanh(eta);
const double coth_err_mult = fabs(eta) + 1.0;
gsl_sf_result rH;
int stat_CF1 = legendre_H3d_CF1_ser(ell, lambda, coth_eta, &rH);
double Hlm1;
double Hl = GSL_SQRT_DBL_MIN;
double Hlp1 = rH.val * Hl;
int lp;
for(lp=ell; lp>0; lp--) {
double root_term_0 = hypot(lambda,lp);
double root_term_1 = hypot(lambda,lp+1.0);
Hlm1 = ((2.0*lp + 1.0)*coth_eta*Hl - root_term_1 * Hlp1)/root_term_0;
Hlp1 = Hl;
Hl = Hlm1;
}
if(fabs(Hl) > fabs(Hlp1)) {
gsl_sf_result H0;
int stat_H0 = gsl_sf_legendre_H3d_0_e(lambda, eta, &H0);
result->val = GSL_SQRT_DBL_MIN/Hl * H0.val;
result->err = GSL_SQRT_DBL_MIN/fabs(Hl) * H0.err;
result->err += fabs(rH.err/rH.val) * (ell+1.0) * coth_err_mult * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_H0, stat_CF1);
}
else {
gsl_sf_result H1;
int stat_H1 = gsl_sf_legendre_H3d_1_e(lambda, eta, &H1);
result->val = GSL_SQRT_DBL_MIN/Hlp1 * H1.val;
result->err = GSL_SQRT_DBL_MIN/fabs(Hlp1) * H1.err;
result->err += fabs(rH.err/rH.val) * (ell+1.0) * coth_err_mult * fabs(result->val);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_H1, stat_CF1);
}
}
}
int
gsl_sf_legendre_H3d_array(const int lmax, const double lambda, const double eta, double * result_array)
{
/* CHECK_POINTER(result_array) */
if(eta < 0.0 || lmax < 0) {
int ell;
for(ell=0; ell<=lmax; ell++) result_array[ell] = 0.0;
GSL_ERROR ("domain error", GSL_EDOM);
}
else if(eta > GSL_LOG_DBL_MAX) {
/* cosh(eta) is too big. */
int ell;
for(ell=0; ell<=lmax; ell++) result_array[ell] = 0.0;
GSL_ERROR ("overflow", GSL_EOVRFLW);
}
else if(lmax == 0) {
gsl_sf_result H0;
int stat = gsl_sf_legendre_H3d_e(0, lambda, eta, &H0);
result_array[0] = H0.val;
return stat;
}
else {
/* Not the most efficient method. But what the hell... it's simple.
*/
gsl_sf_result r_Hlp1;
gsl_sf_result r_Hl;
int stat_lmax = gsl_sf_legendre_H3d_e(lmax, lambda, eta, &r_Hlp1);
int stat_lmaxm1 = gsl_sf_legendre_H3d_e(lmax-1, lambda, eta, &r_Hl);
int stat_max = GSL_ERROR_SELECT_2(stat_lmax, stat_lmaxm1);
const double coth_eta = 1.0/tanh(eta);
int stat_recursion = GSL_SUCCESS;
double Hlp1 = r_Hlp1.val;
double Hl = r_Hl.val;
double Hlm1;
int ell;
result_array[lmax] = Hlp1;
result_array[lmax-1] = Hl;
for(ell=lmax-1; ell>0; ell--) {
double root_term_0 = hypot(lambda,ell);
double root_term_1 = hypot(lambda,ell+1.0);
Hlm1 = ((2.0*ell + 1.0)*coth_eta*Hl - root_term_1 * Hlp1)/root_term_0;
result_array[ell-1] = Hlm1;
if(!(Hlm1 < GSL_DBL_MAX)) stat_recursion = GSL_EOVRFLW;
Hlp1 = Hl;
Hl = Hlm1;
}
return GSL_ERROR_SELECT_2(stat_recursion, stat_max);
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_legendre_H3d_0(const double lambda, const double eta)
{
EVAL_RESULT(gsl_sf_legendre_H3d_0_e(lambda, eta, &result));
}
double gsl_sf_legendre_H3d_1(const double lambda, const double eta)
{
EVAL_RESULT(gsl_sf_legendre_H3d_1_e(lambda, eta, &result));
}
double gsl_sf_legendre_H3d(const int l, const double lambda, const double eta)
{
EVAL_RESULT(gsl_sf_legendre_H3d_e(l, lambda, eta, &result));
}