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Diffstat (limited to 'gsl-1.9/specfunc/legendre_poly.c')
-rw-r--r-- | gsl-1.9/specfunc/legendre_poly.c | 783 |
1 files changed, 783 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/legendre_poly.c b/gsl-1.9/specfunc/legendre_poly.c new file mode 100644 index 0000000..211a101 --- /dev/null +++ b/gsl-1.9/specfunc/legendre_poly.c @@ -0,0 +1,783 @@ +/* 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)); +} + |