| /* specfunc/legendre_poly.c |
| * |
| * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002 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_exp.h> |
| #include <gsl/gsl_sf_gamma.h> |
| #include <gsl/gsl_sf_log.h> |
| #include <gsl/gsl_sf_pow_int.h> |
| #include <gsl/gsl_sf_legendre.h> |
| |
| #include "error.h" |
| |
| |
| |
| /* Calculate P_m^m(x) from the analytic result: |
| * P_m^m(x) = (-1)^m (2m-1)!! (1-x^2)^(m/2) , m > 0 |
| * = 1 , m = 0 |
| */ |
| static double legendre_Pmm(int m, double x) |
| { |
| if(m == 0) |
| { |
| return 1.0; |
| } |
| else |
| { |
| double p_mm = 1.0; |
| double root_factor = sqrt(1.0-x)*sqrt(1.0+x); |
| double fact_coeff = 1.0; |
| int i; |
| for(i=1; i<=m; i++) |
| { |
| p_mm *= -fact_coeff * root_factor; |
| fact_coeff += 2.0; |
| } |
| return p_mm; |
| } |
| } |
| |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| int |
| gsl_sf_legendre_P1_e(double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| { |
| result->val = x; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| } |
| |
| int |
| gsl_sf_legendre_P2_e(double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| { |
| result->val = 0.5*(3.0*x*x - 1.0); |
| result->err = GSL_DBL_EPSILON * (fabs(3.0*x*x) + 1.0); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| int |
| gsl_sf_legendre_P3_e(double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| { |
| result->val = 0.5*x*(5.0*x*x - 3.0); |
| result->err = GSL_DBL_EPSILON * (fabs(result->val) + 0.5 * fabs(x) * (fabs(5.0*x*x) + 3.0)); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Pl_e(const int l, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(l < 0 || x < -1.0 || x > 1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(l == 0) { |
| result->val = 1.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(l == 1) { |
| result->val = x; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(l == 2) { |
| result->val = 0.5 * (3.0*x*x - 1.0); |
| result->err = GSL_DBL_EPSILON * (fabs(3.0*x*x) + 1.0); |
| /*result->err = 3.0 * GSL_DBL_EPSILON * fabs(result->val); |
| removed this old bogus estimate [GJ] |
| */ |
| return GSL_SUCCESS; |
| } |
| else if(x == 1.0) { |
| result->val = 1.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(x == -1.0) { |
| result->val = ( GSL_IS_ODD(l) ? -1.0 : 1.0 ); |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(l < 100000) { |
| /* upward recurrence: l P_l = (2l-1) z P_{l-1} - (l-1) P_{l-2} */ |
| |
| double p_ellm2 = 1.0; /* P_0(x) */ |
| double p_ellm1 = x; /* P_1(x) */ |
| double p_ell = p_ellm1; |
| |
| double e_ellm2 = GSL_DBL_EPSILON; |
| double e_ellm1 = fabs(x)*GSL_DBL_EPSILON; |
| double e_ell = e_ellm1; |
| |
| int ell; |
| |
| for(ell=2; ell <= l; ell++){ |
| p_ell = (x*(2*ell-1)*p_ellm1 - (ell-1)*p_ellm2) / ell; |
| p_ellm2 = p_ellm1; |
| p_ellm1 = p_ell; |
| |
| e_ell = 0.5*(fabs(x)*(2*ell-1.0) * e_ellm1 + (ell-1.0)*e_ellm2)/ell; |
| e_ellm2 = e_ellm1; |
| e_ellm1 = e_ell; |
| } |
| |
| result->val = p_ell; |
| result->err = e_ell + l*fabs(p_ell)*GSL_DBL_EPSILON; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* Asymptotic expansion. |
| * [Olver, p. 473] |
| */ |
| double u = l + 0.5; |
| double th = acos(x); |
| gsl_sf_result J0; |
| gsl_sf_result Jm1; |
| int stat_J0 = gsl_sf_bessel_J0_e(u*th, &J0); |
| int stat_Jm1 = gsl_sf_bessel_Jn_e(-1, u*th, &Jm1); |
| double pre; |
| double B00; |
| double c1; |
| |
| /* 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); |
| } |
| |
| c1 = th/u * B00; |
| |
| result->val = pre * (J0.val + c1 * Jm1.val); |
| result->err = pre * (J0.err + fabs(c1) * Jm1.err); |
| result->err += GSL_SQRT_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_2(stat_J0, stat_Jm1); |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Pl_array(const int lmax, const double x, double * result_array) |
| { |
| /* CHECK_POINTER(result_array) */ |
| |
| if(lmax < 0 || x < -1.0 || x > 1.0) { |
| GSL_ERROR ("domain error", GSL_EDOM); |
| } |
| else if(lmax == 0) { |
| result_array[0] = 1.0; |
| return GSL_SUCCESS; |
| } |
| else if(lmax == 1) { |
| result_array[0] = 1.0; |
| result_array[1] = x; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* upward recurrence: l P_l = (2l-1) z P_{l-1} - (l-1) P_{l-2} */ |
| |
| double p_ellm2 = 1.0; /* P_0(x) */ |
| double p_ellm1 = x; /* P_1(x) */ |
| double p_ell = p_ellm1; |
| int ell; |
| |
| result_array[0] = 1.0; |
| result_array[1] = x; |
| |
| for(ell=2; ell <= lmax; ell++){ |
| p_ell = (x*(2*ell-1)*p_ellm1 - (ell-1)*p_ellm2) / ell; |
| p_ellm2 = p_ellm1; |
| p_ellm1 = p_ell; |
| result_array[ell] = p_ell; |
| } |
| |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Pl_deriv_array(const int lmax, const double x, double * result_array, double * result_deriv_array) |
| { |
| int stat_array = gsl_sf_legendre_Pl_array(lmax, x, result_array); |
| |
| if(lmax >= 0) result_deriv_array[0] = 0.0; |
| if(lmax >= 1) result_deriv_array[1] = 1.0; |
| |
| if(stat_array == GSL_SUCCESS) |
| { |
| int ell; |
| |
| if(fabs(x - 1.0)*(lmax+1.0)*(lmax+1.0) < GSL_SQRT_DBL_EPSILON) |
| { |
| /* x is near 1 */ |
| for(ell = 2; ell <= lmax; ell++) |
| { |
| const double pre = 0.5 * ell * (ell+1.0); |
| result_deriv_array[ell] = pre * (1.0 - 0.25 * (1.0-x) * (ell+2.0)*(ell-1.0)); |
| } |
| } |
| else if(fabs(x + 1.0)*(lmax+1.0)*(lmax+1.0) < GSL_SQRT_DBL_EPSILON) |
| { |
| /* x is near -1 */ |
| for(ell = 2; ell <= lmax; ell++) |
| { |
| const double sgn = ( GSL_IS_ODD(ell) ? 1.0 : -1.0 ); /* derivative is odd in x for even ell */ |
| const double pre = sgn * 0.5 * ell * (ell+1.0); |
| result_deriv_array[ell] = pre * (1.0 - 0.25 * (1.0+x) * (ell+2.0)*(ell-1.0)); |
| } |
| } |
| else |
| { |
| const double diff_a = 1.0 + x; |
| const double diff_b = 1.0 - x; |
| for(ell = 2; ell <= lmax; ell++) |
| { |
| result_deriv_array[ell] = - ell * (x * result_array[ell] - result_array[ell-1]) / (diff_a * diff_b); |
| } |
| } |
| |
| return GSL_SUCCESS; |
| } |
| else |
| { |
| return stat_array; |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Plm_e(const int l, const int m, const double x, gsl_sf_result * result) |
| { |
| /* If l is large and m is large, then we have to worry |
| * about overflow. Calculate an approximate exponent which |
| * measures the normalization of this thing. |
| */ |
| const double dif = l-m; |
| const double sum = l+m; |
| const double t_d = ( dif == 0.0 ? 0.0 : 0.5 * dif * (log(dif)-1.0) ); |
| const double t_s = ( dif == 0.0 ? 0.0 : 0.5 * sum * (log(sum)-1.0) ); |
| const double exp_check = 0.5 * log(2.0*l+1.0) + t_d - t_s; |
| |
| /* CHECK_POINTER(result) */ |
| |
| if(m < 0 || l < m || x < -1.0 || x > 1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(exp_check < GSL_LOG_DBL_MIN + 10.0){ |
| /* Bail out. */ |
| OVERFLOW_ERROR(result); |
| } |
| else { |
| /* Account for the error due to the |
| * representation of 1-x. |
| */ |
| const double err_amp = 1.0 / (GSL_DBL_EPSILON + fabs(1.0-fabs(x))); |
| |
| /* P_m^m(x) and P_{m+1}^m(x) */ |
| double p_mm = legendre_Pmm(m, x); |
| double p_mmp1 = x * (2*m + 1) * p_mm; |
| |
| if(l == m){ |
| result->val = p_mm; |
| result->err = err_amp * 2.0 * GSL_DBL_EPSILON * fabs(p_mm); |
| return GSL_SUCCESS; |
| } |
| else if(l == m + 1) { |
| result->val = p_mmp1; |
| result->err = err_amp * 2.0 * GSL_DBL_EPSILON * fabs(p_mmp1); |
| return GSL_SUCCESS; |
| } |
| else{ |
| /* upward recurrence: (l-m) P(l,m) = (2l-1) z P(l-1,m) - (l+m-1) P(l-2,m) |
| * start at P(m,m), P(m+1,m) |
| */ |
| |
| double p_ellm2 = p_mm; |
| double p_ellm1 = p_mmp1; |
| double p_ell = 0.0; |
| int ell; |
| |
| for(ell=m+2; ell <= l; ell++){ |
| p_ell = (x*(2*ell-1)*p_ellm1 - (ell+m-1)*p_ellm2) / (ell-m); |
| p_ellm2 = p_ellm1; |
| p_ellm1 = p_ell; |
| } |
| |
| result->val = p_ell; |
| result->err = err_amp * (0.5*(l-m) + 1.0) * GSL_DBL_EPSILON * fabs(p_ell); |
| |
| return GSL_SUCCESS; |
| } |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Plm_array(const int lmax, const int m, const double x, double * result_array) |
| { |
| /* If l is large and m is large, then we have to worry |
| * about overflow. Calculate an approximate exponent which |
| * measures the normalization of this thing. |
| */ |
| const double dif = lmax-m; |
| const double sum = lmax+m; |
| const double t_d = ( dif == 0.0 ? 0.0 : 0.5 * dif * (log(dif)-1.0) ); |
| const double t_s = ( dif == 0.0 ? 0.0 : 0.5 * sum * (log(sum)-1.0) ); |
| const double exp_check = 0.5 * log(2.0*lmax+1.0) + t_d - t_s; |
| |
| /* CHECK_POINTER(result_array) */ |
| |
| if(m < 0 || lmax < m || x < -1.0 || x > 1.0) { |
| GSL_ERROR ("domain error", GSL_EDOM); |
| } |
| else if(m > 0 && (x == 1.0 || x == -1.0)) { |
| int ell; |
| for(ell=m; ell<=lmax; ell++) result_array[ell-m] = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(exp_check < GSL_LOG_DBL_MIN + 10.0){ |
| /* Bail out. */ |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| else { |
| double p_mm = legendre_Pmm(m, x); |
| double p_mmp1 = x * (2.0*m + 1.0) * p_mm; |
| |
| if(lmax == m){ |
| result_array[0] = p_mm; |
| return GSL_SUCCESS; |
| } |
| else if(lmax == m + 1) { |
| result_array[0] = p_mm; |
| result_array[1] = p_mmp1; |
| return GSL_SUCCESS; |
| } |
| else { |
| double p_ellm2 = p_mm; |
| double p_ellm1 = p_mmp1; |
| double p_ell = 0.0; |
| int ell; |
| |
| result_array[0] = p_mm; |
| result_array[1] = p_mmp1; |
| |
| for(ell=m+2; ell <= lmax; ell++){ |
| p_ell = (x*(2.0*ell-1.0)*p_ellm1 - (ell+m-1)*p_ellm2) / (ell-m); |
| p_ellm2 = p_ellm1; |
| p_ellm1 = p_ell; |
| result_array[ell-m] = p_ell; |
| } |
| |
| return GSL_SUCCESS; |
| } |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_Plm_deriv_array( |
| const int lmax, const int m, const double x, |
| double * result_array, |
| double * result_deriv_array) |
| { |
| if(m < 0 || m > lmax) |
| { |
| GSL_ERROR("m < 0 or m > lmax", GSL_EDOM); |
| } |
| else if(m == 0) |
| { |
| /* It is better to do m=0 this way, so we can more easily |
| * trap the divergent case which can occur when m == 1. |
| */ |
| return gsl_sf_legendre_Pl_deriv_array(lmax, x, result_array, result_deriv_array); |
| } |
| else |
| { |
| int stat_array = gsl_sf_legendre_Plm_array(lmax, m, x, result_array); |
| |
| if(stat_array == GSL_SUCCESS) |
| { |
| int ell; |
| |
| if(m == 1 && (1.0 - fabs(x) < GSL_DBL_EPSILON)) |
| { |
| /* This divergence is real and comes from the cusp-like |
| * behaviour for m = 1. For example, P[1,1] = - Sqrt[1-x^2]. |
| */ |
| GSL_ERROR("divergence near |x| = 1.0 since m = 1", GSL_EOVRFLW); |
| } |
| else if(m == 2 && (1.0 - fabs(x) < GSL_DBL_EPSILON)) |
| { |
| /* m = 2 gives a finite nonzero result for |x| near 1 */ |
| if(fabs(x - 1.0) < GSL_DBL_EPSILON) |
| { |
| for(ell = m; ell <= lmax; ell++) result_deriv_array[ell-m] = -0.25 * x * (ell - 1.0)*ell*(ell+1.0)*(ell+2.0); |
| } |
| else if(fabs(x + 1.0) < GSL_DBL_EPSILON) |
| { |
| for(ell = m; ell <= lmax; ell++) |
| { |
| const double sgn = ( GSL_IS_ODD(ell) ? 1.0 : -1.0 ); |
| result_deriv_array[ell-m] = -0.25 * sgn * x * (ell - 1.0)*ell*(ell+1.0)*(ell+2.0); |
| } |
| } |
| return GSL_SUCCESS; |
| } |
| else |
| { |
| /* m > 2 is easier to deal with since the endpoints always vanish */ |
| if(1.0 - fabs(x) < GSL_DBL_EPSILON) |
| { |
| for(ell = m; ell <= lmax; ell++) result_deriv_array[ell-m] = 0.0; |
| return GSL_SUCCESS; |
| } |
| else |
| { |
| const double diff_a = 1.0 + x; |
| const double diff_b = 1.0 - x; |
| result_deriv_array[0] = - m * x / (diff_a * diff_b) * result_array[0]; |
| if(lmax-m >= 1) result_deriv_array[1] = (2.0 * m + 1.0) * (x * result_deriv_array[0] + result_array[0]); |
| for(ell = m+2; ell <= lmax; ell++) |
| { |
| result_deriv_array[ell-m] = - (ell * x * result_array[ell-m] - (ell+m) * result_array[ell-1-m]) / (diff_a * diff_b); |
| } |
| return GSL_SUCCESS; |
| } |
| } |
| } |
| else |
| { |
| return stat_array; |
| } |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_sphPlm_e(const int l, int m, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(m < 0 || l < m || x < -1.0 || x > 1.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(m == 0) { |
| gsl_sf_result P; |
| int stat_P = gsl_sf_legendre_Pl_e(l, x, &P); |
| double pre = sqrt((2.0*l + 1.0)/(4.0*M_PI)); |
| result->val = pre * P.val; |
| result->err = pre * P.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat_P; |
| } |
| else if(x == 1.0 || x == -1.0) { |
| /* m > 0 here */ |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| /* m > 0 and |x| < 1 here */ |
| |
| /* Starting value for recursion. |
| * Y_m^m(x) = sqrt( (2m+1)/(4pi m) gamma(m+1/2)/gamma(m) ) (-1)^m (1-x^2)^(m/2) / pi^(1/4) |
| */ |
| gsl_sf_result lncirc; |
| gsl_sf_result lnpoch; |
| double lnpre_val; |
| double lnpre_err; |
| gsl_sf_result ex_pre; |
| double sr; |
| const double sgn = ( GSL_IS_ODD(m) ? -1.0 : 1.0); |
| const double y_mmp1_factor = x * sqrt(2.0*m + 3.0); |
| double y_mm, y_mm_err; |
| double y_mmp1, y_mmp1_err; |
| gsl_sf_log_1plusx_e(-x*x, &lncirc); |
| gsl_sf_lnpoch_e(m, 0.5, &lnpoch); /* Gamma(m+1/2)/Gamma(m) */ |
| lnpre_val = -0.25*M_LNPI + 0.5 * (lnpoch.val + m*lncirc.val); |
| lnpre_err = 0.25*M_LNPI*GSL_DBL_EPSILON + 0.5 * (lnpoch.err + fabs(m)*lncirc.err); |
| /* Compute exp(ln_pre) with error term, avoiding call to gsl_sf_exp_err BJG */ |
| ex_pre.val = exp(lnpre_val); |
| ex_pre.err = 2.0*(sinh(lnpre_err) + GSL_DBL_EPSILON)*ex_pre.val; |
| sr = sqrt((2.0+1.0/m)/(4.0*M_PI)); |
| y_mm = sgn * sr * ex_pre.val; |
| y_mm_err = 2.0 * GSL_DBL_EPSILON * fabs(y_mm) + sr * ex_pre.err; |
| y_mm_err *= 1.0 + 1.0/(GSL_DBL_EPSILON + fabs(1.0-x)); |
| y_mmp1 = y_mmp1_factor * y_mm; |
| y_mmp1_err=fabs(y_mmp1_factor) * y_mm_err; |
| |
| if(l == m){ |
| result->val = y_mm; |
| result->err = y_mm_err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(y_mm); |
| return GSL_SUCCESS; |
| } |
| else if(l == m + 1) { |
| result->val = y_mmp1; |
| result->err = y_mmp1_err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(y_mmp1); |
| return GSL_SUCCESS; |
| } |
| else{ |
| double y_ell = 0.0; |
| double y_ell_err; |
| int ell; |
| |
| /* Compute Y_l^m, l > m+1, upward recursion on l. */ |
| for(ell=m+2; ell <= l; ell++){ |
| const double rat1 = (double)(ell-m)/(double)(ell+m); |
| const double rat2 = (ell-m-1.0)/(ell+m-1.0); |
| const double factor1 = sqrt(rat1*(2.0*ell+1.0)*(2.0*ell-1.0)); |
| const double factor2 = sqrt(rat1*rat2*(2.0*ell+1.0)/(2.0*ell-3.0)); |
| y_ell = (x*y_mmp1*factor1 - (ell+m-1.0)*y_mm*factor2) / (ell-m); |
| y_mm = y_mmp1; |
| y_mmp1 = y_ell; |
| |
| y_ell_err = 0.5*(fabs(x*factor1)*y_mmp1_err + fabs((ell+m-1.0)*factor2)*y_mm_err) / fabs(ell-m); |
| y_mm_err = y_mmp1_err; |
| y_mmp1_err = y_ell_err; |
| } |
| |
| result->val = y_ell; |
| result->err = y_ell_err + (0.5*(l-m) + 1.0) * GSL_DBL_EPSILON * fabs(y_ell); |
| |
| return GSL_SUCCESS; |
| } |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_sphPlm_array(const int lmax, int m, const double x, double * result_array) |
| { |
| /* CHECK_POINTER(result_array) */ |
| |
| if(m < 0 || lmax < m || x < -1.0 || x > 1.0) { |
| GSL_ERROR ("error", GSL_EDOM); |
| } |
| else if(m > 0 && (x == 1.0 || x == -1.0)) { |
| int ell; |
| for(ell=m; ell<=lmax; ell++) result_array[ell-m] = 0.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| double y_mm; |
| double y_mmp1; |
| |
| if(m == 0) { |
| y_mm = 0.5/M_SQRTPI; /* Y00 = 1/sqrt(4pi) */ |
| y_mmp1 = x * M_SQRT3 * y_mm; |
| } |
| else { |
| /* |x| < 1 here */ |
| |
| gsl_sf_result lncirc; |
| gsl_sf_result lnpoch; |
| double lnpre; |
| const double sgn = ( GSL_IS_ODD(m) ? -1.0 : 1.0); |
| gsl_sf_log_1plusx_e(-x*x, &lncirc); |
| gsl_sf_lnpoch_e(m, 0.5, &lnpoch); /* Gamma(m+1/2)/Gamma(m) */ |
| lnpre = -0.25*M_LNPI + 0.5 * (lnpoch.val + m*lncirc.val); |
| y_mm = sqrt((2.0+1.0/m)/(4.0*M_PI)) * sgn * exp(lnpre); |
| y_mmp1 = x * sqrt(2.0*m + 3.0) * y_mm; |
| } |
| |
| if(lmax == m){ |
| result_array[0] = y_mm; |
| return GSL_SUCCESS; |
| } |
| else if(lmax == m + 1) { |
| result_array[0] = y_mm; |
| result_array[1] = y_mmp1; |
| return GSL_SUCCESS; |
| } |
| else{ |
| double y_ell; |
| int ell; |
| |
| result_array[0] = y_mm; |
| result_array[1] = y_mmp1; |
| |
| /* Compute Y_l^m, l > m+1, upward recursion on l. */ |
| for(ell=m+2; ell <= lmax; ell++){ |
| const double rat1 = (double)(ell-m)/(double)(ell+m); |
| const double rat2 = (ell-m-1.0)/(ell+m-1.0); |
| const double factor1 = sqrt(rat1*(2*ell+1)*(2*ell-1)); |
| const double factor2 = sqrt(rat1*rat2*(2*ell+1)/(2*ell-3)); |
| y_ell = (x*y_mmp1*factor1 - (ell+m-1)*y_mm*factor2) / (ell-m); |
| y_mm = y_mmp1; |
| y_mmp1 = y_ell; |
| result_array[ell-m] = y_ell; |
| } |
| } |
| |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int |
| gsl_sf_legendre_sphPlm_deriv_array( |
| const int lmax, const int m, const double x, |
| double * result_array, |
| double * result_deriv_array) |
| { |
| if(m < 0 || lmax < m || x < -1.0 || x > 1.0) |
| { |
| GSL_ERROR ("domain", GSL_EDOM); |
| } |
| else if(m == 0) |
| { |
| /* m = 0 is easy to trap */ |
| const int stat_array = gsl_sf_legendre_Pl_deriv_array(lmax, x, result_array, result_deriv_array); |
| int ell; |
| for(ell = 0; ell <= lmax; ell++) |
| { |
| const double prefactor = sqrt((2.0 * ell + 1.0)/(4.0*M_PI)); |
| result_array[ell] *= prefactor; |
| result_deriv_array[ell] *= prefactor; |
| } |
| return stat_array; |
| } |
| else if(m == 1) |
| { |
| /* Trapping m = 1 is necessary because of the possible divergence. |
| * Recall that this divergence is handled properly in ..._Plm_deriv_array(), |
| * and the scaling factor is not large for small m, so we just scale. |
| */ |
| const int stat_array = gsl_sf_legendre_Plm_deriv_array(lmax, m, x, result_array, result_deriv_array); |
| int ell; |
| for(ell = 1; ell <= lmax; ell++) |
| { |
| const double prefactor = sqrt((2.0 * ell + 1.0)/(ell + 1.0) / (4.0*M_PI*ell)); |
| result_array[ell-1] *= prefactor; |
| result_deriv_array[ell-1] *= prefactor; |
| } |
| return stat_array; |
| } |
| else |
| { |
| /* as for the derivative of P_lm, everything is regular for m >= 2 */ |
| |
| int stat_array = gsl_sf_legendre_sphPlm_array(lmax, m, x, result_array); |
| |
| if(stat_array == GSL_SUCCESS) |
| { |
| int ell; |
| |
| if(1.0 - fabs(x) < GSL_DBL_EPSILON) |
| { |
| for(ell = m; ell <= lmax; ell++) result_deriv_array[ell-m] = 0.0; |
| return GSL_SUCCESS; |
| } |
| else |
| { |
| const double diff_a = 1.0 + x; |
| const double diff_b = 1.0 - x; |
| result_deriv_array[0] = - m * x / (diff_a * diff_b) * result_array[0]; |
| if(lmax-m >= 1) result_deriv_array[1] = sqrt(2.0 * m + 3.0) * (x * result_deriv_array[0] + result_array[0]); |
| for(ell = m+2; ell <= lmax; ell++) |
| { |
| const double c1 = sqrt(((2.0*ell+1.0)/(2.0*ell-1.0)) * ((double)(ell-m)/(double)(ell+m))); |
| result_deriv_array[ell-m] = - (ell * x * result_array[ell-m] - c1 * (ell+m) * result_array[ell-1-m]) / (diff_a * diff_b); |
| } |
| return GSL_SUCCESS; |
| } |
| } |
| else |
| { |
| return stat_array; |
| } |
| } |
| } |
| |
| |
| #ifndef HIDE_INLINE_STATIC |
| int |
| gsl_sf_legendre_array_size(const int lmax, const int m) |
| { |
| return lmax-m+1; |
| } |
| #endif |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_legendre_P1(const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_P1_e(x, &result)); |
| } |
| |
| double gsl_sf_legendre_P2(const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_P2_e(x, &result)); |
| } |
| |
| double gsl_sf_legendre_P3(const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_P3_e(x, &result)); |
| } |
| |
| double gsl_sf_legendre_Pl(const int l, const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_Pl_e(l, x, &result)); |
| } |
| |
| double gsl_sf_legendre_Plm(const int l, const int m, const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_Plm_e(l, m, x, &result)); |
| } |
| |
| double gsl_sf_legendre_sphPlm(const int l, const int m, const double x) |
| { |
| EVAL_RESULT(gsl_sf_legendre_sphPlm_e(l, m, x, &result)); |
| } |
| |
