/* 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 3 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_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 = 0.0;

      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;

    }

  }

}

#ifndef GSL_DISABLE_DEPRECATED

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_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;

    }

  }

}

int

gsl_sf_legendre_array_size(const int lmax, const int m)

{

  return lmax-m+1;

}

#endif /* !GSL_DISABLE_DEPRECATED */

/*-*-*-*-*-*-*-*-*-* 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));

}