/* specfunc/bessel_k.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000 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_gamma.h>

#include <gsl/gsl_sf_bessel.h>

#include "error.h"

#include "check.h"

#include "bessel.h"

/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/

/* [Abramowitz+Stegun, 10.2.4 + 10.2.6]

 * with lmax=15, precision ~ 15D for x < 3

 *

 * assumes l >= 1

 */

static int bessel_kl_scaled_small_x(int l, const double x, gsl_sf_result * result)

{

  gsl_sf_result num_fact;

  double den  = gsl_sf_pow_int(x, l+1);

  int stat_df = gsl_sf_doublefact_e((unsigned int) (2*l-1), &num_fact);

  if(stat_df != GSL_SUCCESS || den == 0.0) {

    OVERFLOW_ERROR(result);

  }

  else {

    const int lmax = 50;

    gsl_sf_result ipos_term;

    double ineg_term;

    double sgn = (GSL_IS_ODD(l) ? -1.0 : 1.0);

    double ex  = exp(x);

    double t = 0.5*x*x;

    double sum = 1.0;

    double t_coeff = 1.0;

    double t_power = 1.0;

    double delta;

    int stat_il;

    int i;

    for(i=1; i

      t_coeff /= i*(2*(i-l) - 1);

      t_power *= t;

      delta = t_power*t_coeff;

      sum += delta;

      if(fabs(delta/sum) < GSL_DBL_EPSILON) break;

    }

    stat_il = gsl_sf_bessel_il_scaled_e(l, x, &ipos_term);

    ineg_term =  sgn * num_fact.val/den * sum;

    result->val = -sgn * 0.5*M_PI * (ex*ipos_term.val - ineg_term);

    result->val *= ex;

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

    return stat_il;

  }

}

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

int gsl_sf_bessel_k0_scaled_e(const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0) {

    DOMAIN_ERROR(result);

  }

  else {

    result->val = M_PI/(2.0*x);

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

    CHECK_UNDERFLOW(result);

    return GSL_SUCCESS;

  }

}

int gsl_sf_bessel_k1_scaled_e(const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x < (M_SQRTPI+1.0)/(M_SQRT2*GSL_SQRT_DBL_MAX)) {

    OVERFLOW_ERROR(result);

  }

  else {

    result->val = M_PI/(2.0*x) * (1.0 + 1.0/x);

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

    CHECK_UNDERFLOW(result);

    return GSL_SUCCESS;

  }

}

int gsl_sf_bessel_k2_scaled_e(const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(x < 2.0/GSL_ROOT3_DBL_MAX) {

    OVERFLOW_ERROR(result);

  }

  else {

    result->val = M_PI/(2.0*x) * (1.0 + 3.0/x * (1.0 + 1.0/x));

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

    CHECK_UNDERFLOW(result);

    return GSL_SUCCESS;

  }

}

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

{

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

    DOMAIN_ERROR(result);

  }

  else if(l == 0) {

    return gsl_sf_bessel_k0_scaled_e(x, result);

  }

  else if(l == 1) {

    return gsl_sf_bessel_k1_scaled_e(x, result);

  }

  else if(l == 2) {

    return gsl_sf_bessel_k2_scaled_e(x, result);

  }

  else if(x < 3.0) {

    return bessel_kl_scaled_small_x(l, x, result);

  }

  else if(GSL_ROOT3_DBL_EPSILON * x > (l*l + l + 1)) {

    int status = gsl_sf_bessel_Knu_scaled_asympx_e(l + 0.5, x, result);

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

    result->val *= pre;

    result->err *= pre;

    return status;

  }

  else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < GSL_ROOT3_DBL_EPSILON) {

    int status = gsl_sf_bessel_Knu_scaled_asymp_unif_e(l + 0.5, x, result);

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

    result->val *= pre;

    result->err *= pre;

    return status;

  }

  else {

    /* recurse upward */

    gsl_sf_result r_bk;

    gsl_sf_result r_bkm;

    int stat_1 = gsl_sf_bessel_k1_scaled_e(x, &r_bk);

    int stat_0 = gsl_sf_bessel_k0_scaled_e(x, &r_bkm);

    double bkp;

    double bk  = r_bk.val;

    double bkm = r_bkm.val;

    int j;

    for(j=1; j

      bkp = (2*j+1)/x*bk + bkm;

      bkm = bk;

      bk  = bkp;

    }

    result->val  = bk;

    result->err  = fabs(bk) * (fabs(r_bk.err/r_bk.val) + fabs(r_bkm.err/r_bkm.val));

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

    return GSL_ERROR_SELECT_2(stat_1, stat_0);

  }

}

int 

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

{

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

    GSL_ERROR("domain error", GSL_EDOM);

  } else if (lmax == 0) {

    gsl_sf_result result;

    int stat = gsl_sf_bessel_k0_scaled_e(x, &result);

    result_array[0] = result.val;

    return stat;

  } else {

    int ell;

    double kellp1, kell, kellm1;

    gsl_sf_result r_kell;

    gsl_sf_result r_kellm1;

    gsl_sf_bessel_k1_scaled_e(x, &r_kell);

    gsl_sf_bessel_k0_scaled_e(x, &r_kellm1);

    kell   = r_kell.val;

    kellm1 = r_kellm1.val;

    result_array[0] = kellm1;

    result_array[1] = kell;

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

      kellp1 = (2*ell+1)/x * kell + kellm1;

      result_array[ell+1] = kellp1;

      kellm1 = kell;

      kell   = kellp1;

    }

    return GSL_SUCCESS;

  }

}

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

#include "eval.h"

double gsl_sf_bessel_k0_scaled(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_k0_scaled_e(x, &result));

}

double gsl_sf_bessel_k1_scaled(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_k1_scaled_e(x, &result));

}

double gsl_sf_bessel_k2_scaled(const double x)

{

  EVAL_RESULT(gsl_sf_bessel_k2_scaled_e(x, &result));

}

double gsl_sf_bessel_kl_scaled(const int l, const double x)

{

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

}