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/* specfunc/legendre_Qn.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_bessel.h>
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#include <gsl/gsl_sf_elementary.h>
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#include <gsl/gsl_sf_exp.h>
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#include <gsl/gsl_sf_pow_int.h>
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#include <gsl/gsl_sf_legendre.h>
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#include "error.h"
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/* Evaluate f_{ell+1}/f_ell
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* f_ell := Q^{b}_{a+ell}(x)
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* x > 1
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*/
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static
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int
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legendreQ_CF1_xgt1(int ell, double a, double b, double x, double * 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 = ell + 1.0 + a + b;
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double b1 = (2.0*(ell+1.0+a) + 1.0) * x;
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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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double lna;
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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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lna = ell + n + a;
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an = b*b - lna*lna;
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bn = (2.0*lna + 1.0) * x;
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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 = 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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/* Uniform asymptotic for Q_l(x).
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* Assumes x > -1.0 and x != 1.0.
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* Discards second order and higher terms.
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*/
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static
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int
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legendre_Ql_asymp_unif(const double ell, const double x, gsl_sf_result * result)
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{
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if(x < 1.0) {
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double u = ell + 0.5;
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double th = acos(x);
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gsl_sf_result Y0, Y1;
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int stat_Y0, stat_Y1;
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int stat_m;
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double pre;
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double B00;
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double sum;
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/* B00 = 1/8 (1 - th cot(th) / th^2
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* pre = sqrt(th/sin(th))
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*/
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if(th < GSL_ROOT4_DBL_EPSILON) {
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B00 = (1.0 + th*th/15.0)/24.0;
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pre = 1.0 + th*th/12.0;
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}
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else {
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double sin_th = sqrt(1.0 - x*x);
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double cot_th = x / sin_th;
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B00 = 1.0/8.0 * (1.0 - th * cot_th) / (th*th);
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pre = sqrt(th/sin_th);
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}
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stat_Y0 = gsl_sf_bessel_Y0_e(u*th, &Y0;;
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stat_Y1 = gsl_sf_bessel_Y1_e(u*th, &Y1;;
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sum = -0.5*M_PI * (Y0.val + th/u * Y1.val * B00);
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stat_m = gsl_sf_multiply_e(pre, sum, result);
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result->err += 0.5*M_PI * fabs(pre) * (Y0.err + fabs(th/u*B00)*Y1.err);
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result->err += GSL_DBL_EPSILON * fabs(result->val);
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return GSL_ERROR_SELECT_3(stat_m, stat_Y0, stat_Y1);
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}
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else {
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double u = ell + 0.5;
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double xi = acosh(x);
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gsl_sf_result K0_scaled, K1_scaled;
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int stat_K0, stat_K1;
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int stat_e;
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double pre;
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double B00;
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double sum;
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/* B00 = -1/8 (1 - xi coth(xi) / xi^2
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* pre = sqrt(xi/sinh(xi))
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*/
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if(xi < GSL_ROOT4_DBL_EPSILON) {
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B00 = (1.0-xi*xi/15.0)/24.0;
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pre = 1.0 - xi*xi/12.0;
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}
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else {
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double sinh_xi = sqrt(x*x - 1.0);
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double coth_xi = x / sinh_xi;
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B00 = -1.0/8.0 * (1.0 - xi * coth_xi) / (xi*xi);
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pre = sqrt(xi/sinh_xi);
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}
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stat_K0 = gsl_sf_bessel_K0_scaled_e(u*xi, &K0_scaled);
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stat_K1 = gsl_sf_bessel_K1_scaled_e(u*xi, &K1_scaled);
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sum = K0_scaled.val - xi/u * K1_scaled.val * B00;
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stat_e = gsl_sf_exp_mult_e(-u*xi, pre * sum, result);
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result->err = GSL_DBL_EPSILON * fabs(result->val) * fabs(u*xi);
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result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
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return GSL_ERROR_SELECT_3(stat_e, stat_K0, stat_K1);
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}
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}
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|
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/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
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int
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gsl_sf_legendre_Q0_e(const double x, gsl_sf_result * result)
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{
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/* CHECK_POINTER(result) */
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if(x <= -1.0 || x == 1.0) {
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|
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DOMAIN_ERROR(result);
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|
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}
|
|
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else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */
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const double c3 = 1.0/3.0;
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const double c5 = 1.0/5.0;
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const double c7 = 1.0/7.0;
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const double c9 = 1.0/9.0;
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const double c11 = 1.0/11.0;
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const double y = x * x;
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const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11))));
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|
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result->val = x * series;
|
|
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result->err = 2.0 * GSL_DBL_EPSILON * fabs(x);
|
|
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67cb25 |
return GSL_SUCCESS;
|
|
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}
|
|
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else if(x < 1.0) {
|
|
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result->val = 0.5 * log((1.0+x)/(1.0-x));
|
|
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67cb25 |
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
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67cb25 |
return GSL_SUCCESS;
|
|
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67cb25 |
}
|
|
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else if(x < 10.0) {
|
|
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result->val = 0.5 * log((x+1.0)/(x-1.0));
|
|
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67cb25 |
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
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67cb25 |
return GSL_SUCCESS;
|
|
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67cb25 |
}
|
|
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67cb25 |
else if(x*GSL_DBL_MIN < 2.0) {
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|
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const double y = 1.0/(x*x);
|
|
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67cb25 |
const double c1 = 1.0/3.0;
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|
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67cb25 |
const double c2 = 1.0/5.0;
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const double c3 = 1.0/7.0;
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|
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const double c4 = 1.0/9.0;
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const double c5 = 1.0/11.0;
|
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const double c6 = 1.0/13.0;
|
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const double c7 = 1.0/15.0;
|
|
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result->val = (1.0/x) * (1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*c7)))))));
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result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
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return GSL_SUCCESS;
|
|
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}
|
|
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else {
|
|
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UNDERFLOW_ERROR(result);
|
|
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}
|
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}
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|
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int
|
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gsl_sf_legendre_Q1_e(const double x, gsl_sf_result * result)
|
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67cb25 |
{
|
|
Packit |
67cb25 |
/* CHECK_POINTER(result) */
|
|
Packit |
67cb25 |
|
|
Packit |
67cb25 |
if(x <= -1.0 || x == 1.0) {
|
|
Packit |
67cb25 |
DOMAIN_ERROR(result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */
|
|
Packit |
67cb25 |
const double c3 = 1.0/3.0;
|
|
Packit |
67cb25 |
const double c5 = 1.0/5.0;
|
|
Packit |
67cb25 |
const double c7 = 1.0/7.0;
|
|
Packit |
67cb25 |
const double c9 = 1.0/9.0;
|
|
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67cb25 |
const double c11 = 1.0/11.0;
|
|
Packit |
67cb25 |
const double y = x * x;
|
|
Packit |
67cb25 |
const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11))));
|
|
Packit |
67cb25 |
result->val = x * x * series - 1.0;
|
|
Packit |
67cb25 |
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(x < 1.0){
|
|
Packit |
67cb25 |
result->val = 0.5 * x * (log((1.0+x)/(1.0-x))) - 1.0;
|
|
Packit |
67cb25 |
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(x < 6.0) {
|
|
Packit |
67cb25 |
result->val = 0.5 * x * log((x+1.0)/(x-1.0)) - 1.0;
|
|
Packit |
67cb25 |
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else if(x*GSL_SQRT_DBL_MIN < 0.99/M_SQRT3) {
|
|
Packit |
67cb25 |
const double y = 1/(x*x);
|
|
Packit |
67cb25 |
const double c1 = 3.0/5.0;
|
|
Packit |
67cb25 |
const double c2 = 3.0/7.0;
|
|
Packit |
67cb25 |
const double c3 = 3.0/9.0;
|
|
Packit |
67cb25 |
const double c4 = 3.0/11.0;
|
|
Packit |
67cb25 |
const double c5 = 3.0/13.0;
|
|
Packit |
67cb25 |
const double c6 = 3.0/15.0;
|
|
Packit |
67cb25 |
const double c7 = 3.0/17.0;
|
|
Packit |
67cb25 |
const double c8 = 3.0/19.0;
|
|
Packit |
67cb25 |
const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*(c7 + y*c8)))))));
|
|
Packit |
67cb25 |
result->val = sum / (3.0*x*x);
|
|
Packit |
67cb25 |
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
|
|
Packit |
67cb25 |
return GSL_SUCCESS;
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
else {
|
|
Packit |
67cb25 |
UNDERFLOW_ERROR(result);
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
}
|
|
Packit |
67cb25 |
|
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int
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gsl_sf_legendre_Ql_e(const int l, const double x, gsl_sf_result * result)
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{
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/* CHECK_POINTER(result) */
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if(x <= -1.0 || x == 1.0 || l < 0) {
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DOMAIN_ERROR(result);
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}
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else if(l == 0) {
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return gsl_sf_legendre_Q0_e(x, result);
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}
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else if(l == 1) {
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return gsl_sf_legendre_Q1_e(x, result);
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}
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else if(l > 100000) {
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return legendre_Ql_asymp_unif(l, x, result);
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}
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else if(x < 1.0){
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/* Forward recurrence.
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*/
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gsl_sf_result Q0, Q1;
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int stat_Q0 = gsl_sf_legendre_Q0_e(x, &Q0;;
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int stat_Q1 = gsl_sf_legendre_Q1_e(x, &Q1;;
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double Qellm1 = Q0.val;
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double Qell = Q1.val;
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double Qellp1;
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int ell;
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for(ell=1; ell
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Qellp1 = (x*(2.0*ell + 1.0) * Qell - ell * Qellm1) / (ell + 1.0);
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Qellm1 = Qell;
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Qell = Qellp1;
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}
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result->val = Qell;
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result->err = GSL_DBL_EPSILON * l * fabs(result->val);
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return GSL_ERROR_SELECT_2(stat_Q0, stat_Q1);
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}
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else {
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/* x > 1.0 */
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double rat;
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int stat_CF1 = legendreQ_CF1_xgt1(l, 0.0, 0.0, x, &rat);
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int stat_Q;
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double Qellp1 = rat * GSL_SQRT_DBL_MIN;
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double Qell = GSL_SQRT_DBL_MIN;
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double Qellm1;
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int ell;
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for(ell=l; ell>0; ell--) {
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Qellm1 = (x * (2.0*ell + 1.0) * Qell - (ell+1.0) * Qellp1) / ell;
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Qellp1 = Qell;
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Qell = Qellm1;
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}
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if(fabs(Qell) > fabs(Qellp1)) {
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gsl_sf_result Q0;
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stat_Q = gsl_sf_legendre_Q0_e(x, &Q0;;
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result->val = GSL_SQRT_DBL_MIN * Q0.val / Qell;
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result->err = l * GSL_DBL_EPSILON * fabs(result->val);
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}
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else {
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gsl_sf_result Q1;
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stat_Q = gsl_sf_legendre_Q1_e(x, &Q1;;
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result->val = GSL_SQRT_DBL_MIN * Q1.val / Qellp1;
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result->err = l * GSL_DBL_EPSILON * fabs(result->val);
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}
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return GSL_ERROR_SELECT_2(stat_Q, stat_CF1);
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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_Q0(const double x)
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{
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EVAL_RESULT(gsl_sf_legendre_Q0_e(x, &result));
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}
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double gsl_sf_legendre_Q1(const double x)
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{
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EVAL_RESULT(gsl_sf_legendre_Q1_e(x, &result));
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}
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double gsl_sf_legendre_Ql(const int l, const double x)
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{
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EVAL_RESULT(gsl_sf_legendre_Ql_e(l, x, &result));
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}
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