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+/* 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));
+}
+
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