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/* specfunc/legendre_H3d.c
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*
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* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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/* Author: G. Jungman */
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#include <config.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_errno.h>
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#include <gsl/gsl_sf_exp.h>
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#include <gsl/gsl_sf_gamma.h>
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#include <gsl/gsl_sf_trig.h>
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#include <gsl/gsl_sf_legendre.h>
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#include "error.h"
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#include "legendre.h"
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/* See [Abbott+Schaefer, Ap.J. 308, 546 (1986)] for
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* enough details to follow what is happening here.
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*/
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/* Logarithm of normalization factor, Log[N(ell,lambda)].
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* N(ell,lambda) = Product[ lambda^2 + n^2, {n,0,ell} ]
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* = |Gamma(ell + 1 + I lambda)|^2 lambda sinh(Pi lambda) / Pi
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* Assumes ell >= 0.
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*/
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static
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int
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legendre_H3d_lnnorm(const int ell, const double lambda, double * result)
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{
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double abs_lam = fabs(lambda);
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if(abs_lam == 0.0) {
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*result = 0.0;
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GSL_ERROR ("error", GSL_EDOM);
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}
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else if(lambda > (ell + 1.0)/GSL_ROOT3_DBL_EPSILON) {
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/* There is a cancellation between the sinh(Pi lambda)
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* term and the log(gamma(ell + 1 + i lambda) in the
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* result below, so we show some care and save some digits.
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* Note that the above guarantees that lambda is large,
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* since ell >= 0. We use Stirling and a simple expansion
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* of sinh.
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*/
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double rat = (ell+1.0)/lambda;
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double ln_lam2ell2 = 2.0*log(lambda) + log(1.0 + rat*rat);
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double lg_corrected = -2.0*(ell+1.0) + M_LNPI + (ell+0.5)*ln_lam2ell2 + 1.0/(288.0*lambda*lambda);
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double angle_terms = lambda * 2.0 * rat * (1.0 - rat*rat/3.0);
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*result = log(abs_lam) + lg_corrected + angle_terms - M_LNPI;
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return GSL_SUCCESS;
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}
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else {
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gsl_sf_result lg_r;
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gsl_sf_result lg_theta;
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gsl_sf_result ln_sinh;
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gsl_sf_lngamma_complex_e(ell+1.0, lambda, &lg_r, &lg_theta);
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gsl_sf_lnsinh_e(M_PI * abs_lam, &ln_sinh);
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*result = log(abs_lam) + ln_sinh.val + 2.0*lg_r.val - M_LNPI;
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return GSL_SUCCESS;
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}
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}
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/* Calculate series for small eta*lambda.
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* Assumes eta > 0, lambda != 0.
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*
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* This is just the defining hypergeometric for the Legendre function.
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*
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* P^{mu}_{-1/2 + I lam}(z) = 1/Gamma(l+3/2) ((z+1)/(z-1)^(mu/2)
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* 2F1(1/2 - I lam, 1/2 + I lam; l+3/2; (1-z)/2)
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* We use
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* z = cosh(eta)
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* (z-1)/2 = sinh^2(eta/2)
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*
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* And recall
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* H3d = sqrt(Pi Norm /(2 lam^2 sinh(eta))) P^{-l-1/2}_{-1/2 + I lam}(cosh(eta))
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*/
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static
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int
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legendre_H3d_series(const int ell, const double lambda, const double eta,
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gsl_sf_result * result)
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{
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const int nmax = 5000;
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const double shheta = sinh(0.5*eta);
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const double ln_zp1 = M_LN2 + log(1.0 + shheta*shheta);
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const double ln_zm1 = M_LN2 + 2.0*log(shheta);
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const double zeta = -shheta*shheta;
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gsl_sf_result lg_lp32;
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double term = 1.0;
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double sum = 1.0;
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double sum_err = 0.0;
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gsl_sf_result lnsheta;
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double lnN;
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double lnpre_val, lnpre_err, lnprepow;
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int stat_e;
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int n;
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gsl_sf_lngamma_e(ell + 3.0/2.0, &lg_lp32);
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gsl_sf_lnsinh_e(eta, &lnsheta);
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legendre_H3d_lnnorm(ell, lambda, &lnN;;
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lnprepow = 0.5*(ell + 0.5) * (ln_zm1 - ln_zp1);
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lnpre_val = lnprepow + 0.5*(lnN + M_LNPI - M_LN2 - lnsheta.val) - lg_lp32.val - log(fabs(lambda));
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lnpre_err = lnsheta.err + lg_lp32.err + GSL_DBL_EPSILON * fabs(lnpre_val);
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lnpre_err += 2.0*GSL_DBL_EPSILON * (fabs(lnN) + M_LNPI + M_LN2);
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lnpre_err += 2.0*GSL_DBL_EPSILON * (0.5*(ell + 0.5) * (fabs(ln_zm1) + fabs(ln_zp1)));
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for(n=1; n
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double aR = n - 0.5;
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term *= (aR*aR + lambda*lambda)*zeta/(ell + n + 0.5)/n;
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sum += term;
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sum_err += 2.0*GSL_DBL_EPSILON*fabs(term);
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if(fabs(term/sum) < 2.0 * GSL_DBL_EPSILON) break;
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}
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stat_e = gsl_sf_exp_mult_err_e(lnpre_val, lnpre_err, sum, fabs(term)+sum_err, result);
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return GSL_ERROR_SELECT_2(stat_e, (n==nmax ? GSL_EMAXITER : GSL_SUCCESS));
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}
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/* Evaluate legendre_H3d(ell+1)/legendre_H3d(ell)
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* by continued fraction.
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*/
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#if 0
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static
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int
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legendre_H3d_CF1(const int ell, const double lambda, const double coth_eta,
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gsl_sf_result * result)
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{
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const double RECUR_BIG = GSL_SQRT_DBL_MAX;
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const int maxiter = 5000;
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int n = 1;
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double Anm2 = 1.0;
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double Bnm2 = 0.0;
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double Anm1 = 0.0;
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double Bnm1 = 1.0;
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double a1 = hypot(lambda, ell+1.0);
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double b1 = (2.0*ell + 3.0) * coth_eta;
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double An = b1*Anm1 + a1*Anm2;
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double Bn = b1*Bnm1 + a1*Bnm2;
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double an, bn;
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double fn = An/Bn;
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while(n < maxiter) {
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double old_fn;
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double del;
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n++;
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Anm2 = Anm1;
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Bnm2 = Bnm1;
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Anm1 = An;
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Bnm1 = Bn;
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an = -(lambda*lambda + ((double)ell + n)*((double)ell + n));
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bn = (2.0*ell + 2.0*n + 1.0) * coth_eta;
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An = bn*Anm1 + an*Anm2;
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Bn = bn*Bnm1 + an*Bnm2;
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if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {
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An /= RECUR_BIG;
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Bn /= RECUR_BIG;
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Anm1 /= RECUR_BIG;
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Bnm1 /= RECUR_BIG;
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Anm2 /= RECUR_BIG;
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Bnm2 /= RECUR_BIG;
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}
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old_fn = fn;
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fn = An/Bn;
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del = old_fn/fn;
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if(fabs(del - 1.0) < 4.0*GSL_DBL_EPSILON) break;
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}
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result->val = fn;
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result->err = 2.0 * GSL_DBL_EPSILON * (sqrt(n)+1.0) * fabs(fn);
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if(n >= maxiter)
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GSL_ERROR ("error", GSL_EMAXITER);
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else
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return GSL_SUCCESS;
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}
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#endif /* 0 */
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/* Evaluate legendre_H3d(ell+1)/legendre_H3d(ell)
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* by continued fraction. Use the Gautschi (Euler)
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* equivalent series.
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*/
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/* FIXME: Maybe we have to worry about this. The a_k are
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* not positive and there can be a blow-up. It happened
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* for J_nu once or twice. Then we should probably use
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* the method above.
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*/
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static
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int
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legendre_H3d_CF1_ser(const int ell, const double lambda, const double coth_eta,
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gsl_sf_result * result)
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{
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const double pre = hypot(lambda, ell+1.0)/((2.0*ell+3)*coth_eta);
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const int maxk = 20000;
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double tk = 1.0;
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double sum = 1.0;
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double rhok = 0.0;
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double sum_err = 0.0;
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int k;
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for(k=1; k
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double tlk = (2.0*ell + 1.0 + 2.0*k);
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double l1k = (ell + 1.0 + k);
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double ak = -(lambda*lambda + l1k*l1k)/(tlk*(tlk+2.0)*coth_eta*coth_eta);
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rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
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tk *= rhok;
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sum += tk;
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sum_err += 2.0 * GSL_DBL_EPSILON * k * fabs(tk);
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if(fabs(tk/sum) < GSL_DBL_EPSILON) break;
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}
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result->val = pre * sum;
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result->err = fabs(pre * tk);
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result->err += fabs(pre * sum_err);
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result->err += 4.0 * GSL_DBL_EPSILON * fabs(result->val);
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if(k >= maxk)
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GSL_ERROR ("error", GSL_EMAXITER);
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else
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return GSL_SUCCESS;
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}
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|
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/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
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|
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int
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gsl_sf_legendre_H3d_0_e(const double lambda, const double eta, gsl_sf_result * result)
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{
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/* CHECK_POINTER(result) */
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if(eta < 0.0) {
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DOMAIN_ERROR(result);
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}
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else if(eta == 0.0 || lambda == 0.0) {
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result->val = 1.0;
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result->err = 0.0;
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return GSL_SUCCESS;
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}
|
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else {
|
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const double lam_eta = lambda * eta;
|
|
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gsl_sf_result s;
|
|
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gsl_sf_sin_err_e(lam_eta, 2.0*GSL_DBL_EPSILON * fabs(lam_eta), &s);
|
|
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if(eta > -0.5*GSL_LOG_DBL_EPSILON) {
|
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double f = 2.0 / lambda * exp(-eta);
|
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result->val = f * s.val;
|
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result->err = fabs(f * s.val) * (fabs(eta) + 1.0) * GSL_DBL_EPSILON;
|
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result->err += fabs(f) * s.err;
|
|
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result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
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}
|
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else {
|
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double f = 1.0/(lambda*sinh(eta));
|
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result->val = f * s.val;
|
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67cb25 |
result->err = fabs(f * s.val) * (fabs(eta) + 1.0) * GSL_DBL_EPSILON;
|
|
Packit |
67cb25 |
result->err += fabs(f) * s.err;
|
|
Packit |
67cb25 |
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
int
|
|
Packit |
67cb25 |
gsl_sf_legendre_H3d_1_e(const double lambda, const double eta, gsl_sf_result * result)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
const double xi = fabs(eta*lambda);
|
|
Packit |
67cb25 |
const double lsq = lambda*lambda;
|
|
Packit |
67cb25 |
const double lsqp1 = lsq + 1.0;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
/* CHECK_POINTER(result) */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if(eta < 0.0) {
|
|
Packit |
67cb25 |
DOMAIN_ERROR(result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(eta == 0.0 || lambda == 0.0) {
|
|
Packit |
67cb25 |
result->val = 0.0;
|
|
Packit |
67cb25 |
result->err = 0.0;
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(xi < GSL_ROOT5_DBL_EPSILON && eta < GSL_ROOT5_DBL_EPSILON) {
|
|
Packit |
67cb25 |
double etasq = eta*eta;
|
|
Packit |
67cb25 |
double xisq = xi*xi;
|
|
Packit |
67cb25 |
double term1 = (etasq + xisq)/3.0;
|
|
Packit |
67cb25 |
double term2 = -(2.0*etasq*etasq + 5.0*etasq*xisq + 3.0*xisq*xisq)/90.0;
|
|
Packit |
67cb25 |
double sinh_term = 1.0 - eta*eta/6.0 * (1.0 - 7.0/60.0*eta*eta);
|
|
Packit |
67cb25 |
double pre = sinh_term/sqrt(lsqp1) / eta;
|
|
Packit |
67cb25 |
result->val = pre * (term1 + term2);
|
|
Packit |
67cb25 |
result->err = pre * GSL_DBL_EPSILON * (fabs(term1) + fabs(term2));
|
|
Packit |
67cb25 |
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
double sin_term; /* Sin(xi)/xi */
|
|
Packit |
67cb25 |
double cos_term; /* Cos(xi) */
|
|
Packit |
67cb25 |
double coth_term; /* eta/Tanh(eta) */
|
|
Packit |
67cb25 |
double sinh_term; /* eta/Sinh(eta) */
|
|
Packit |
67cb25 |
double sin_term_err;
|
|
Packit |
67cb25 |
double cos_term_err;
|
|
Packit |
67cb25 |
double t1;
|
|
Packit |
67cb25 |
double pre_val;
|
|
Packit |
67cb25 |
double pre_err;
|
|
Packit |
67cb25 |
double term1;
|
|
Packit |
67cb25 |
double term2;
|
|
Packit |
67cb25 |
if(xi < GSL_ROOT5_DBL_EPSILON) {
|
|
Packit |
67cb25 |
sin_term = 1.0 - xi*xi/6.0 * (1.0 - xi*xi/20.0);
|
|
Packit |
67cb25 |
cos_term = 1.0 - 0.5*xi*xi * (1.0 - xi*xi/12.0);
|
|
Packit |
67cb25 |
sin_term_err = GSL_DBL_EPSILON;
|
|
Packit |
67cb25 |
cos_term_err = GSL_DBL_EPSILON;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
gsl_sf_result sin_xi_result;
|
|
Packit |
67cb25 |
gsl_sf_result cos_xi_result;
|
|
Packit |
67cb25 |
gsl_sf_sin_e(xi, &sin_xi_result);
|
|
Packit |
67cb25 |
gsl_sf_cos_e(xi, &cos_xi_result);
|
|
Packit |
67cb25 |
sin_term = sin_xi_result.val/xi;
|
|
Packit |
67cb25 |
cos_term = cos_xi_result.val;
|
|
Packit |
67cb25 |
sin_term_err = sin_xi_result.err/fabs(xi);
|
|
Packit |
67cb25 |
cos_term_err = cos_xi_result.err;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
if(eta < GSL_ROOT5_DBL_EPSILON) {
|
|
Packit |
67cb25 |
coth_term = 1.0 + eta*eta/3.0 * (1.0 - eta*eta/15.0);
|
|
Packit |
67cb25 |
sinh_term = 1.0 - eta*eta/6.0 * (1.0 - 7.0/60.0*eta*eta);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
coth_term = eta/tanh(eta);
|
|
Packit |
67cb25 |
sinh_term = eta/sinh(eta);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
t1 = sqrt(lsqp1) * eta;
|
|
Packit |
67cb25 |
pre_val = sinh_term/t1;
|
|
Packit |
67cb25 |
pre_err = 2.0 * GSL_DBL_EPSILON * fabs(pre_val);
|
|
Packit |
67cb25 |
term1 = sin_term*coth_term;
|
|
Packit |
67cb25 |
term2 = cos_term;
|
|
Packit |
67cb25 |
result->val = pre_val * (term1 - term2);
|
|
Packit |
67cb25 |
result->err = pre_err * fabs(term1 - term2);
|
|
Packit |
67cb25 |
result->err += pre_val * (sin_term_err * coth_term + cos_term_err);
|
|
Packit |
67cb25 |
result->err += pre_val * fabs(term1-term2) * (fabs(eta) + 1.0) * GSL_DBL_EPSILON;
|
|
Packit |
67cb25 |
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
int
|
|
Packit |
67cb25 |
gsl_sf_legendre_H3d_e(const int ell, const double lambda, const double eta,
|
|
Packit |
67cb25 |
gsl_sf_result * result)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
const double abs_lam = fabs(lambda);
|
|
Packit |
67cb25 |
const double lsq = abs_lam*abs_lam;
|
|
Packit |
67cb25 |
const double xi = abs_lam * eta;
|
|
Packit |
67cb25 |
const double cosh_eta = cosh(eta);
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
/* CHECK_POINTER(result) */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if(eta < 0.0) {
|
|
Packit |
67cb25 |
DOMAIN_ERROR(result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(eta > GSL_LOG_DBL_MAX) {
|
|
Packit |
67cb25 |
/* cosh(eta) is too big. */
|
|
Packit |
67cb25 |
OVERFLOW_ERROR(result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(ell == 0) {
|
|
Packit |
67cb25 |
return gsl_sf_legendre_H3d_0_e(lambda, eta, result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(ell == 1) {
|
|
Packit |
67cb25 |
return gsl_sf_legendre_H3d_1_e(lambda, eta, result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(eta == 0.0) {
|
|
Packit |
67cb25 |
result->val = 0.0;
|
|
Packit |
67cb25 |
result->err = 0.0;
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(xi < 1.0) {
|
|
Packit |
67cb25 |
return legendre_H3d_series(ell, lambda, eta, result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if((ell*ell+lsq)/sqrt(1.0+lsq)/(cosh_eta*cosh_eta) < 5.0*GSL_ROOT3_DBL_EPSILON) {
|
|
Packit |
67cb25 |
/* Large argument.
|
|
Packit |
67cb25 |
*/
|
|
Packit |
67cb25 |
gsl_sf_result P;
|
|
Packit |
67cb25 |
double lm;
|
|
Packit |
67cb25 |
int stat_P = gsl_sf_conicalP_large_x_e(-ell-0.5, lambda, cosh_eta, &P, &lm);
|
|
Packit |
67cb25 |
if(P.val == 0.0) {
|
|
Packit |
67cb25 |
result->val = 0.0;
|
|
Packit |
67cb25 |
result->err = 0.0;
|
|
Packit |
67cb25 |
return stat_P;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
double lnN;
|
|
Packit |
67cb25 |
gsl_sf_result lnsh;
|
|
Packit |
67cb25 |
double ln_abslam;
|
|
Packit |
67cb25 |
double lnpre_val, lnpre_err;
|
|
Packit |
67cb25 |
int stat_e;
|
|
Packit |
67cb25 |
gsl_sf_lnsinh_e(eta, &lnsh;;
|
|
Packit |
67cb25 |
legendre_H3d_lnnorm(ell, lambda, &lnN;;
|
|
Packit |
67cb25 |
ln_abslam = log(abs_lam);
|
|
Packit |
67cb25 |
lnpre_val = 0.5*(M_LNPI + lnN - M_LN2 - lnsh.val) - ln_abslam;
|
|
Packit |
67cb25 |
lnpre_err = lnsh.err;
|
|
Packit |
67cb25 |
lnpre_err += 2.0 * GSL_DBL_EPSILON * (0.5*(M_LNPI + M_LN2 + fabs(lnN)) + fabs(ln_abslam));
|
|
Packit |
67cb25 |
lnpre_err += 2.0 * GSL_DBL_EPSILON * fabs(lnpre_val);
|
|
Packit |
67cb25 |
stat_e = gsl_sf_exp_mult_err_e(lnpre_val + lm, lnpre_err, P.val, P.err, result);
|
|
Packit |
67cb25 |
return GSL_ERROR_SELECT_2(stat_e, stat_P);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(abs_lam > 1000.0*ell*ell) {
|
|
Packit |
67cb25 |
/* Large degree.
|
|
Packit |
67cb25 |
*/
|
|
Packit |
67cb25 |
gsl_sf_result P;
|
|
Packit |
67cb25 |
double lm;
|
|
Packit |
67cb25 |
int stat_P = gsl_sf_conicalP_xgt1_neg_mu_largetau_e(ell+0.5,
|
|
Packit |
67cb25 |
lambda,
|
|
Packit |
67cb25 |
cosh_eta, eta,
|
|
Packit |
67cb25 |
&P, &lm);
|
|
Packit |
67cb25 |
if(P.val == 0.0) {
|
|
Packit |
67cb25 |
result->val = 0.0;
|
|
Packit |
67cb25 |
result->err = 0.0;
|
|
Packit |
67cb25 |
return stat_P;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
double lnN;
|
|
Packit |
67cb25 |
gsl_sf_result lnsh;
|
|
Packit |
67cb25 |
double ln_abslam;
|
|
Packit |
67cb25 |
double lnpre_val, lnpre_err;
|
|
Packit |
67cb25 |
int stat_e;
|
|
Packit |
67cb25 |
gsl_sf_lnsinh_e(eta, &lnsh;;
|
|
Packit |
67cb25 |
legendre_H3d_lnnorm(ell, lambda, &lnN;;
|
|
Packit |
67cb25 |
ln_abslam = log(abs_lam);
|
|
Packit |
67cb25 |
lnpre_val = 0.5*(M_LNPI + lnN - M_LN2 - lnsh.val) - ln_abslam;
|
|
Packit |
67cb25 |
lnpre_err = lnsh.err;
|
|
Packit |
67cb25 |
lnpre_err += GSL_DBL_EPSILON * (0.5*(M_LNPI + M_LN2 + fabs(lnN)) + fabs(ln_abslam));
|
|
Packit |
67cb25 |
lnpre_err += 2.0 * GSL_DBL_EPSILON * fabs(lnpre_val);
|
|
Packit |
67cb25 |
stat_e = gsl_sf_exp_mult_err_e(lnpre_val + lm, lnpre_err, P.val, P.err, result);
|
|
Packit |
67cb25 |
return GSL_ERROR_SELECT_2(stat_e, stat_P);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
/* Backward recurrence.
|
|
Packit |
67cb25 |
*/
|
|
Packit |
67cb25 |
const double coth_eta = 1.0/tanh(eta);
|
|
Packit |
67cb25 |
const double coth_err_mult = fabs(eta) + 1.0;
|
|
Packit |
67cb25 |
gsl_sf_result rH;
|
|
Packit |
67cb25 |
int stat_CF1 = legendre_H3d_CF1_ser(ell, lambda, coth_eta, &rH);
|
|
Packit |
67cb25 |
double Hlm1;
|
|
Packit |
67cb25 |
double Hl = GSL_SQRT_DBL_MIN;
|
|
Packit |
67cb25 |
double Hlp1 = rH.val * Hl;
|
|
Packit |
67cb25 |
int lp;
|
|
Packit |
67cb25 |
for(lp=ell; lp>0; lp--) {
|
|
Packit |
67cb25 |
double root_term_0 = hypot(lambda,lp);
|
|
Packit |
67cb25 |
double root_term_1 = hypot(lambda,lp+1.0);
|
|
Packit |
67cb25 |
Hlm1 = ((2.0*lp + 1.0)*coth_eta*Hl - root_term_1 * Hlp1)/root_term_0;
|
|
Packit |
67cb25 |
Hlp1 = Hl;
|
|
Packit |
67cb25 |
Hl = Hlm1;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if(fabs(Hl) > fabs(Hlp1)) {
|
|
Packit |
67cb25 |
gsl_sf_result H0;
|
|
Packit |
67cb25 |
int stat_H0 = gsl_sf_legendre_H3d_0_e(lambda, eta, &H0;;
|
|
Packit |
67cb25 |
result->val = GSL_SQRT_DBL_MIN/Hl * H0.val;
|
|
Packit |
67cb25 |
result->err = GSL_SQRT_DBL_MIN/fabs(Hl) * H0.err;
|
|
Packit |
67cb25 |
result->err += fabs(rH.err/rH.val) * (ell+1.0) * coth_err_mult * fabs(result->val);
|
|
Packit |
67cb25 |
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_ERROR_SELECT_2(stat_H0, stat_CF1);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
gsl_sf_result H1;
|
|
Packit |
67cb25 |
int stat_H1 = gsl_sf_legendre_H3d_1_e(lambda, eta, &H1;;
|
|
Packit |
67cb25 |
result->val = GSL_SQRT_DBL_MIN/Hlp1 * H1.val;
|
|
Packit |
67cb25 |
result->err = GSL_SQRT_DBL_MIN/fabs(Hlp1) * H1.err;
|
|
Packit |
67cb25 |
result->err += fabs(rH.err/rH.val) * (ell+1.0) * coth_err_mult * fabs(result->val);
|
|
Packit |
67cb25 |
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_ERROR_SELECT_2(stat_H1, stat_CF1);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
int
|
|
Packit |
67cb25 |
gsl_sf_legendre_H3d_array(const int lmax, const double lambda, const double eta, double * result_array)
|
|
Packit |
67cb25 |
{
|
|
Packit |
67cb25 |
/* CHECK_POINTER(result_array) */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if(eta < 0.0 || lmax < 0) {
|
|
Packit |
67cb25 |
int ell;
|
|
Packit |
67cb25 |
for(ell=0; ell<=lmax; ell++) result_array[ell] = 0.0;
|
|
Packit |
67cb25 |
GSL_ERROR ("domain error", GSL_EDOM);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(eta > GSL_LOG_DBL_MAX) {
|
|
Packit |
67cb25 |
/* cosh(eta) is too big. */
|
|
Packit |
67cb25 |
int ell;
|
|
Packit |
67cb25 |
for(ell=0; ell<=lmax; ell++) result_array[ell] = 0.0;
|
|
Packit |
67cb25 |
GSL_ERROR ("overflow", GSL_EOVRFLW);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(lmax == 0) {
|
|
Packit |
67cb25 |
gsl_sf_result H0;
|
|
Packit |
67cb25 |
int stat = gsl_sf_legendre_H3d_e(0, lambda, eta, &H0;;
|
|
Packit |
67cb25 |
result_array[0] = H0.val;
|
|
Packit |
67cb25 |
return stat;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
/* Not the most efficient method. But what the hell... it's simple.
|
|
Packit |
67cb25 |
*/
|
|
Packit |
67cb25 |
gsl_sf_result r_Hlp1;
|
|
Packit |
67cb25 |
gsl_sf_result r_Hl;
|
|
Packit |
67cb25 |
int stat_lmax = gsl_sf_legendre_H3d_e(lmax, lambda, eta, &r_Hlp1);
|
|
Packit |
67cb25 |
int stat_lmaxm1 = gsl_sf_legendre_H3d_e(lmax-1, lambda, eta, &r_Hl);
|
|
Packit |
67cb25 |
int stat_max = GSL_ERROR_SELECT_2(stat_lmax, stat_lmaxm1);
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
const double coth_eta = 1.0/tanh(eta);
|
|
Packit |
67cb25 |
int stat_recursion = GSL_SUCCESS;
|
|
Packit |
67cb25 |
double Hlp1 = r_Hlp1.val;
|
|
Packit |
67cb25 |
double Hl = r_Hl.val;
|
|
Packit |
67cb25 |
double Hlm1;
|
|
Packit |
67cb25 |
int ell;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
result_array[lmax] = Hlp1;
|
|
Packit |
67cb25 |
result_array[lmax-1] = Hl;
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
for(ell=lmax-1; ell>0; ell--) {
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double root_term_0 = hypot(lambda,ell);
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double root_term_1 = hypot(lambda,ell+1.0);
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Hlm1 = ((2.0*ell + 1.0)*coth_eta*Hl - root_term_1 * Hlp1)/root_term_0;
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result_array[ell-1] = Hlm1;
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if(!(Hlm1 < GSL_DBL_MAX)) stat_recursion = GSL_EOVRFLW;
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Hlp1 = Hl;
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Hl = Hlm1;
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}
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return GSL_ERROR_SELECT_2(stat_recursion, stat_max);
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}
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}
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/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
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#include "eval.h"
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double gsl_sf_legendre_H3d_0(const double lambda, const double eta)
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{
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EVAL_RESULT(gsl_sf_legendre_H3d_0_e(lambda, eta, &result));
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}
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double gsl_sf_legendre_H3d_1(const double lambda, const double eta)
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{
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EVAL_RESULT(gsl_sf_legendre_H3d_1_e(lambda, eta, &result));
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}
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double gsl_sf_legendre_H3d(const int l, const double lambda, const double eta)
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{
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EVAL_RESULT(gsl_sf_legendre_H3d_e(l, lambda, eta, &result));
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}
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