| /* specfunc/legendre_Qn.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 2 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_bessel.h> |
| #include <gsl/gsl_sf_elementary.h> |
| #include <gsl/gsl_sf_exp.h> |
| #include <gsl/gsl_sf_pow_int.h> |
| #include <gsl/gsl_sf_legendre.h> |
| |
| #include "error.h" |
| |
| /* Evaluate f_{ell+1}/f_ell |
| * f_ell := Q^{b}_{a+ell}(x) |
| * x > 1 |
| */ |
| static |
| int |
| legendreQ_CF1_xgt1(int ell, double a, double b, double x, double * 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 = ell + 1.0 + a + b; |
| double b1 = (2.0*(ell+1.0+a) + 1.0) * x; |
| 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; |
| double lna; |
| n++; |
| Anm2 = Anm1; |
| Bnm2 = Bnm1; |
| Anm1 = An; |
| Bnm1 = Bn; |
| lna = ell + n + a; |
| an = b*b - lna*lna; |
| bn = (2.0*lna + 1.0) * x; |
| 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 = fn; |
| |
| if(n == maxiter) |
| GSL_ERROR ("error", GSL_EMAXITER); |
| else |
| return GSL_SUCCESS; |
| } |
| |
| |
| /* Uniform asymptotic for Q_l(x). |
| * Assumes x > -1.0 and x != 1.0. |
| * Discards second order and higher terms. |
| */ |
| static |
| int |
| legendre_Ql_asymp_unif(const double ell, const double x, gsl_sf_result * result) |
| { |
| if(x < 1.0) { |
| double u = ell + 0.5; |
| double th = acos(x); |
| gsl_sf_result Y0, Y1; |
| int stat_Y0, stat_Y1; |
| int stat_m; |
| double pre; |
| double B00; |
| double sum; |
| |
| /* B00 = 1/8 (1 - th cot(th) / th^2 |
| * pre = sqrt(th/sin(th)) |
| */ |
| if(th < GSL_ROOT4_DBL_EPSILON) { |
| B00 = (1.0 + th*th/15.0)/24.0; |
| pre = 1.0 + th*th/12.0; |
| } |
| else { |
| double sin_th = sqrt(1.0 - x*x); |
| double cot_th = x / sin_th; |
| B00 = 1.0/8.0 * (1.0 - th * cot_th) / (th*th); |
| pre = sqrt(th/sin_th); |
| } |
| |
| stat_Y0 = gsl_sf_bessel_Y0_e(u*th, &Y0); |
| stat_Y1 = gsl_sf_bessel_Y1_e(u*th, &Y1); |
| |
| sum = -0.5*M_PI * (Y0.val + th/u * Y1.val * B00); |
| |
| stat_m = gsl_sf_multiply_e(pre, sum, result); |
| result->err += 0.5*M_PI * fabs(pre) * (Y0.err + fabs(th/u*B00)*Y1.err); |
| result->err += GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_3(stat_m, stat_Y0, stat_Y1); |
| } |
| else { |
| double u = ell + 0.5; |
| double xi = acosh(x); |
| gsl_sf_result K0_scaled, K1_scaled; |
| int stat_K0, stat_K1; |
| int stat_e; |
| double pre; |
| double B00; |
| double sum; |
| |
| /* B00 = -1/8 (1 - xi coth(xi) / xi^2 |
| * pre = sqrt(xi/sinh(xi)) |
| */ |
| if(xi < GSL_ROOT4_DBL_EPSILON) { |
| B00 = (1.0-xi*xi/15.0)/24.0; |
| pre = 1.0 - xi*xi/12.0; |
| } |
| else { |
| double sinh_xi = sqrt(x*x - 1.0); |
| double coth_xi = x / sinh_xi; |
| B00 = -1.0/8.0 * (1.0 - xi * coth_xi) / (xi*xi); |
| pre = sqrt(xi/sinh_xi); |
| } |
| |
| stat_K0 = gsl_sf_bessel_K0_scaled_e(u*xi, &K0_scaled); |
| stat_K1 = gsl_sf_bessel_K1_scaled_e(u*xi, &K1_scaled); |
| |
| sum = K0_scaled.val - xi/u * K1_scaled.val * B00; |
| |
| stat_e = gsl_sf_exp_mult_e(-u*xi, pre * sum, result); |
| result->err = GSL_DBL_EPSILON * fabs(result->val) * fabs(u*xi); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_3(stat_e, stat_K0, stat_K1); |
| } |
| } |
| |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| int |
| gsl_sf_legendre_Q0_e(const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0 || x == 1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */ |
| const double c3 = 1.0/3.0; |
| const double c5 = 1.0/5.0; |
| const double c7 = 1.0/7.0; |
| const double c9 = 1.0/9.0; |
| const double c11 = 1.0/11.0; |
| const double y = x * x; |
| const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11)))); |
| result->val = x * series; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(x); |
| return GSL_SUCCESS; |
| } |
| else if(x < 1.0) { |
| result->val = 0.5 * log((1.0+x)/(1.0-x)); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(x < 10.0) { |
| result->val = 0.5 * log((x+1.0)/(x-1.0)); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(x*GSL_DBL_MIN < 2.0) { |
| const double y = 1.0/(x*x); |
| const double c1 = 1.0/3.0; |
| const double c2 = 1.0/5.0; |
| const double c3 = 1.0/7.0; |
| const double c4 = 1.0/9.0; |
| const double c5 = 1.0/11.0; |
| const double c6 = 1.0/13.0; |
| const double c7 = 1.0/15.0; |
| result->val = (1.0/x) * (1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*c7))))))); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| UNDERFLOW_ERROR(result); |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Q1_e(const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0 || x == 1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x*x < GSL_ROOT6_DBL_EPSILON) { /* |x| <~ 0.05 */ |
| const double c3 = 1.0/3.0; |
| const double c5 = 1.0/5.0; |
| const double c7 = 1.0/7.0; |
| const double c9 = 1.0/9.0; |
| const double c11 = 1.0/11.0; |
| const double y = x * x; |
| const double series = 1.0 + y*(c3 + y*(c5 + y*(c7 + y*(c9 + y*c11)))); |
| result->val = x * x * series - 1.0; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(x < 1.0){ |
| result->val = 0.5 * x * (log((1.0+x)/(1.0-x))) - 1.0; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(x < 6.0) { |
| result->val = 0.5 * x * log((x+1.0)/(x-1.0)) - 1.0; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(x*GSL_SQRT_DBL_MIN < 0.99/M_SQRT3) { |
| const double y = 1/(x*x); |
| const double c1 = 3.0/5.0; |
| const double c2 = 3.0/7.0; |
| const double c3 = 3.0/9.0; |
| const double c4 = 3.0/11.0; |
| const double c5 = 3.0/13.0; |
| const double c6 = 3.0/15.0; |
| const double c7 = 3.0/17.0; |
| const double c8 = 3.0/19.0; |
| const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*(c7 + y*c8))))))); |
| result->val = sum / (3.0*x*x); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| UNDERFLOW_ERROR(result); |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Ql_e(const int l, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= -1.0 || x == 1.0 || l < 0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(l == 0) { |
| return gsl_sf_legendre_Q0_e(x, result); |
| } |
| else if(l == 1) { |
| return gsl_sf_legendre_Q1_e(x, result); |
| } |
| else if(l > 100000) { |
| return legendre_Ql_asymp_unif(l, x, result); |
| } |
| else if(x < 1.0){ |
| /* Forward recurrence. |
| */ |
| gsl_sf_result Q0, Q1; |
| int stat_Q0 = gsl_sf_legendre_Q0_e(x, &Q0); |
| int stat_Q1 = gsl_sf_legendre_Q1_e(x, &Q1); |
| double Qellm1 = Q0.val; |
| double Qell = Q1.val; |
| double Qellp1; |
| int ell; |
| for(ell=1; ell<l; ell++) { |
| Qellp1 = (x*(2.0*ell + 1.0) * Qell - ell * Qellm1) / (ell + 1.0); |
| Qellm1 = Qell; |
| Qell = Qellp1; |
| } |
| result->val = Qell; |
| result->err = GSL_DBL_EPSILON * l * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_Q0, stat_Q1); |
| } |
| else { |
| /* x > 1.0 */ |
| |
| double rat; |
| int stat_CF1 = legendreQ_CF1_xgt1(l, 0.0, 0.0, x, &rat); |
| int stat_Q; |
| double Qellp1 = rat * GSL_SQRT_DBL_MIN; |
| double Qell = GSL_SQRT_DBL_MIN; |
| double Qellm1; |
| int ell; |
| for(ell=l; ell>0; ell--) { |
| Qellm1 = (x * (2.0*ell + 1.0) * Qell - (ell+1.0) * Qellp1) / ell; |
| Qellp1 = Qell; |
| Qell = Qellm1; |
| } |
| |
| if(fabs(Qell) > fabs(Qellp1)) { |
| gsl_sf_result Q0; |
| stat_Q = gsl_sf_legendre_Q0_e(x, &Q0); |
| result->val = GSL_SQRT_DBL_MIN * Q0.val / Qell; |
| result->err = l * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| else { |
| gsl_sf_result Q1; |
| stat_Q = gsl_sf_legendre_Q1_e(x, &Q1); |
| result->val = GSL_SQRT_DBL_MIN * Q1.val / Qellp1; |
| result->err = l * GSL_DBL_EPSILON * fabs(result->val); |
| } |
| |
| return GSL_ERROR_SELECT_2(stat_Q, stat_CF1); |
| } |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_legendre_Q0(const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_Q0_e(x, &result)); |
| } |
| |
| double gsl_sf_legendre_Q1(const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_Q1_e(x, &result)); |
| } |
| |
| double gsl_sf_legendre_Ql(const int l, const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_Ql_e(l, x, &result)); |
| } |