/* specfunc/bessel_j.c

 * 

 * Copyright (C) 1996,1997,1998,1999,2000,2001,2002,2003 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_pow_int.h>

#include <gsl/gsl_sf_trig.h>

#include <gsl/gsl_sf_bessel.h>

#include "error.h"

#include "bessel.h"

#include "bessel_olver.h"

/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

int gsl_sf_bessel_j0_e(const double x, gsl_sf_result * result)

{

  double ax = fabs(x);

  /* CHECK_POINTER(result) */

  if(ax < 0.5) {

    const double y = x*x;

    const double c1 = -1.0/6.0;

    const double c2 =  1.0/120.0;

    const double c3 = -1.0/5040.0;

    const double c4 =  1.0/362880.0;

    const double c5 = -1.0/39916800.0;

    const double c6 =  1.0/6227020800.0;

    result->val = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*c6)))));

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

  else {

    result->val = sin(x) / x;

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

}

int gsl_sf_bessel_j1_e(const double x, gsl_sf_result * result)

{

  double ax = fabs(x);

  /* CHECK_POINTER(result) */

  if(x == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(ax < 3.1*GSL_DBL_MIN) {

    UNDERFLOW_ERROR(result);

  }

  else if(ax < 0.25) {

    const double y = x*x;

    const double c1 = -1.0/10.0;

    const double c2 =  1.0/280.0;

    const double c3 = -1.0/15120.0;

    const double c4 =  1.0/1330560.0;

    const double c5 = -1.0/172972800.0;

    const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));

    result->val = x/3.0 * sum;

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

  else {

    const double cos_x = cos(x);

    const double sin_x = sin(x);

    result->val  = (sin_x/x - cos_x)/x;

    result->err  = 2.0 * GSL_DBL_EPSILON * (fabs(sin_x/(x*x)) + fabs(cos_x/x));

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

}

int gsl_sf_bessel_j2_e(const double x, gsl_sf_result * result)

{

  double ax = fabs(x);

  /* CHECK_POINTER(result) */

  if(x == 0.0) {

    result->val = 0.0;

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(ax < 4.0*GSL_SQRT_DBL_MIN) {

    UNDERFLOW_ERROR(result);

  }

  else if(ax < 1.3) {

    const double y  = x*x;

    const double c1 = -1.0/14.0;

    const double c2 =  1.0/504.0;

    const double c3 = -1.0/33264.0;

    const double c4 =  1.0/3459456.0;

    const double c5 = -1.0/518918400;

    const double c6 =  1.0/105859353600.0;

    const double c7 = -1.0/28158588057600.0;

    const double c8 =  1.0/9461285587353600.0;

    const double c9 = -1.0/3916972233164390400.0;

    const double sum = 1.0+y*(c1+y*(c2+y*(c3+y*(c4+y*(c5+y*(c6+y*(c7+y*(c8+y*c9))))))));

    result->val = y/15.0 * sum;

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_SUCCESS;

  }

  else {

    /* bug #45730: switch from gsl_sf_{cos,sin} to cos/sin to fix large inputs */

#if 0

    gsl_sf_result cos_result;

    gsl_sf_result sin_result;

    const int stat_cos = gsl_sf_cos_e(x, &cos_result);

    const int stat_sin = gsl_sf_sin_e(x, &sin_result);

    const double cos_x = cos_result.val;

    const double sin_x = sin_result.val;

    const double f = (3.0/(x*x) - 1.0);

    result->val  = (f * sin_x - 3.0*cos_x/x)/x;

    result->err  = fabs(f * sin_result.err/x) + fabs((3.0*cos_result.err/x)/x);

    result->err += 2.0 * GSL_DBL_EPSILON * (fabs(f*sin_x/x) + 3.0*fabs(cos_x/(x*x)));

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_ERROR_SELECT_2(stat_cos, stat_sin);

#else

    const double cos_x = cos(x);

    const double sin_x = sin(x);

    const double f = (3.0/(x*x) - 1.0);

    result->val  = (f * sin_x - 3.0*cos_x/x)/x;

    result->err = 2.0 * GSL_DBL_EPSILON * (fabs(f*sin_x/x) + 3.0*fabs(cos_x/(x*x)));

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    /*return GSL_ERROR_SELECT_2(stat_cos, stat_sin);*/

    return GSL_SUCCESS;

#endif

  }

}

int

gsl_sf_bessel_jl_e(const int l, const double x, gsl_sf_result * result)

{

  if(l < 0 || x < 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x == 0.0) {

    result->val = ( l > 0 ? 0.0 : 1.0 );

    result->err = 0.0;

    return GSL_SUCCESS;

  }

  else if(l == 0) {

    return gsl_sf_bessel_j0_e(x, result);

  }

  else if(l == 1) {

    return gsl_sf_bessel_j1_e(x, result);

  }

  else if(l == 2) {

    return gsl_sf_bessel_j2_e(x, result);

  }

  else if(x*x < 10.0*(l+0.5)/M_E) {

    gsl_sf_result b;

    int status = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, -1, 50, GSL_DBL_EPSILON, &b);

    double pre   = sqrt((0.5*M_PI)/x);

    result->val  = pre * b.val;

    result->err  = pre * b.err;

    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return status;

  }

  else if(GSL_ROOT4_DBL_EPSILON * x > (l*l + l + 1.0)) {

    gsl_sf_result b;

    int status = gsl_sf_bessel_Jnu_asympx_e(l + 0.5, x, &b);

    double pre = sqrt((0.5*M_PI)/x);

    result->val = pre * b.val;

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err;

    return status;

  }

  else if(l > 1.0/GSL_ROOT6_DBL_EPSILON) {

    gsl_sf_result b;

    int status = gsl_sf_bessel_Jnu_asymp_Olver_e(l + 0.5, x, &b);

    double pre = sqrt((0.5*M_PI)/x);

    result->val = pre * b.val;

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err;

    return status;

  }

  else if(x > 1000.0 && x > l*l)

  {

    /* We need this path to avoid feeding large x to CF1 below; */

    gsl_sf_result b;

    int status = gsl_sf_bessel_Jnu_asympx_e(l + 0.5, x, &b);

    double pre = sqrt((0.5*M_PI)/x);

    result->val = pre * b.val;

    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err;

    return status;  

  }

  else {

    double sgn;

    double ratio;

    /* The CF1 call will hit 10000 iterations for x > 10000 + l */

    int stat_CF1 = gsl_sf_bessel_J_CF1(l+0.5, x, &ratio, &sgn);

    const double BESSEL_J_SMALL = GSL_DBL_MIN / GSL_DBL_EPSILON;

    double jellp1 = BESSEL_J_SMALL * ratio;

    double jell   = BESSEL_J_SMALL;

    double jellm1;

    int ell;

    for(ell = l; ell > 0; ell--) {

      jellm1 = -jellp1 + (2*ell + 1)/x * jell;

      jellp1 = jell;

      jell   = jellm1;

    }

    if(fabs(jell) > fabs(jellp1)) {

      gsl_sf_result j0_result;

      int stat_j0  = gsl_sf_bessel_j0_e(x, &j0_result);

      double pre   = BESSEL_J_SMALL / jell;

      result->val  = j0_result.val * pre;

      result->err  = j0_result.err * fabs(pre);

      result->err += 4.0 * GSL_DBL_EPSILON * (0.5*l + 1.0) * fabs(result->val);

      return GSL_ERROR_SELECT_2(stat_j0, stat_CF1);

    }

    else {

      gsl_sf_result j1_result;

      int stat_j1  = gsl_sf_bessel_j1_e(x, &j1_result);

      double pre   = BESSEL_J_SMALL / jellp1;

      result->val  = j1_result.val * pre;

      result->err  = j1_result.err * fabs(pre);

      result->err += 4.0 * GSL_DBL_EPSILON * (0.5*l + 1.0) * fabs(result->val);

      return GSL_ERROR_SELECT_2(stat_j1, stat_CF1);

    }

  }

}

int

gsl_sf_bessel_jl_array(const int lmax, const double x, double * result_array)

{

  /* CHECK_POINTER(result_array) */

  if(lmax < 0 || x < 0.0) {

    int j;

    for(j=0; j<=lmax; j++) result_array[j] = 0.0;

    GSL_ERROR ("error", GSL_EDOM);

  }

  else if(x == 0.0) {

    int j;

    for(j=1; j<=lmax; j++) result_array[j] = 0.0;

    result_array[0] = 1.0;

    return GSL_SUCCESS;

  }

  else {

    gsl_sf_result r_jellp1;

    gsl_sf_result r_jell;

    int stat_0 = gsl_sf_bessel_jl_e(lmax+1, x, &r_jellp1);

    int stat_1 = gsl_sf_bessel_jl_e(lmax,   x, &r_jell);

    double jellp1 = r_jellp1.val;

    double jell   = r_jell.val;

    double jellm1;

    int ell;

    result_array[lmax] = jell;

    for(ell = lmax; ell >= 1; ell--) {

      jellm1 = -jellp1 + (2*ell + 1)/x * jell;

      jellp1 = jell;

      jell   = jellm1;

      result_array[ell-1] = jellm1;

    }

    return GSL_ERROR_SELECT_2(stat_0, stat_1);

  }

}

int gsl_sf_bessel_jl_steed_array(const int lmax, const double x, double * jl_x)

{

  /* CHECK_POINTER(jl_x) */

  if(lmax < 0 || x < 0.0) {

    int j;

    for(j=0; j<=lmax; j++) jl_x[j] = 0.0;

    GSL_ERROR ("error", GSL_EDOM);

  }

  else if(x == 0.0) {

    int j;

    for(j=1; j<=lmax; j++) jl_x[j] = 0.0;

    jl_x[0] = 1.0;

    return GSL_SUCCESS;

  }

  else if(x < 2.0*GSL_ROOT4_DBL_EPSILON) {

    /* first two terms of Taylor series */

    double inv_fact = 1.0;  /* 1/(1 3 5 ... (2l+1)) */

    double x_l      = 1.0;  /* x^l */

    int l;

    for(l=0; l<=lmax; l++) {

      jl_x[l]  = x_l * inv_fact;

      jl_x[l] *= 1.0 - 0.5*x*x/(2.0*l+3.0);

      inv_fact /= 2.0*l+3.0;

      x_l      *= x;

    }

    return GSL_SUCCESS;

  }

  else {

    /* Steed/Barnett algorithm [Comp. Phys. Comm. 21, 297 (1981)] */

    double x_inv = 1.0/x;

    double W = 2.0*x_inv;

    double F = 1.0;

    double FP = (lmax+1.0) * x_inv;

    double B = 2.0*FP + x_inv;

    double end = B + 20000.0*W;

    double D = 1.0/B;

    double del = -D;

    FP += del;

    /* continued fraction */

    do {

      B += W;

      D = 1.0/(B-D);

      del *= (B*D - 1.);

      FP += del;

      if(D < 0.0) F = -F;

      if(B > end) {

        GSL_ERROR ("error", GSL_EMAXITER);

      }

    }

    while(fabs(del) >= fabs(FP) * GSL_DBL_EPSILON);

    FP *= F;

    if(lmax > 0) {

      /* downward recursion */

      double XP2 = FP;

      double PL = lmax * x_inv;

      int L  = lmax;

      int LP;

      jl_x[lmax] = F;

      for(LP = 1; LP<=lmax; LP++) {

        jl_x[L-1] = PL * jl_x[L] + XP2;

        FP = PL*jl_x[L-1] - jl_x[L];

        XP2 = FP;

        PL -= x_inv;

        --L;

      }

      F = jl_x[0];

    }

    /* normalization */

    W = x_inv / hypot(FP, F);

    jl_x[0] = W*F;

    if(lmax > 0) {

      int L;

      for(L=1; L<=lmax; L++) {

        jl_x[L] *= W;

      }

    }

    return GSL_SUCCESS;

  }

}

/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/

#include "eval.h"

double gsl_sf_bessel_j0(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_j0_e(x, &result));

}

double gsl_sf_bessel_j1(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_j1_e(x, &result));

}

double gsl_sf_bessel_j2(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_j2_e(x, &result));

}

double gsl_sf_bessel_jl(const int l, const double x)

{

  EVAL_RESULT(gsl_sf_bessel_jl_e(l, x, &result));

}