/* specfunc/bessel_Knu.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_exp.h>

#include <gsl/gsl_sf_gamma.h>

#include <gsl/gsl_sf_bessel.h>

#include "error.h"

#include "bessel.h"

#include "bessel_temme.h"

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

int

gsl_sf_bessel_Knu_scaled_e(const double nu, const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0 || nu < 0.0) {

    DOMAIN_ERROR(result);

  }

  else {

    gsl_sf_result_e10 result_e10;

    int status = gsl_sf_bessel_Knu_scaled_e10_e(nu, x, &result_e10);

    int status2 = gsl_sf_result_smash_e(&result_e10, result);

    return GSL_ERROR_SELECT_2(status, status2);

  }

}

int

gsl_sf_bessel_Knu_scaled_e10_e(const double nu, const double x, gsl_sf_result_e10 * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0 || nu < 0.0) {

    DOMAIN_ERROR_E10(result);

  }

  else {

    int N = (int)(nu + 0.5);

    double mu = nu - N;      /* -1/2 <= mu <= 1/2 */

    double K_mu, K_mup1, Kp_mu;

    double K_nu, K_nup1, K_num1;

    int n, e10 = 0;

    if(x < 2.0) {

      gsl_sf_bessel_K_scaled_temme(mu, x, &K_mu, &K_mup1, &Kp_mu);

    }

    else {

      gsl_sf_bessel_K_scaled_steed_temme_CF2(mu, x, &K_mu, &K_mup1, &Kp_mu);

    }

    /* recurse forward to obtain K_num1, K_nu */

    K_nu   = K_mu;

    K_nup1 = K_mup1;

    for(n=0; n

      K_num1 = K_nu;

      K_nu   = K_nup1;

      /* rescale the recurrence to avoid overflow */

      if (fabs(K_nu) > GSL_SQRT_DBL_MAX) {

        double p = floor(log(fabs(K_nu))/M_LN10);

        double factor = pow(10.0, p);

        K_num1 /= factor;

        K_nu /= factor;

        e10 += p;

      };

      K_nup1 = 2.0*(mu+n+1)/x * K_nu + K_num1;

    }

    result->val = K_nu;

    result->err = 2.0 * GSL_DBL_EPSILON * (N + 4.0) * fabs(result->val);

    result->e10 = e10;

    return GSL_SUCCESS;

  }

}

int

gsl_sf_bessel_Knu_e(const double nu, const double x, gsl_sf_result * result)

{

  gsl_sf_result b;

  int stat_K = gsl_sf_bessel_Knu_scaled_e(nu, x, &b);

  int stat_e = gsl_sf_exp_mult_err_e(-x, 0.0, b.val, b.err, result);

  return GSL_ERROR_SELECT_2(stat_e, stat_K);

}

int

gsl_sf_bessel_lnKnu_e(const double nu, const double x, gsl_sf_result * result)

{

  /* CHECK_POINTER(result) */

  if(x <= 0.0 || nu < 0.0) {

    DOMAIN_ERROR(result);

  }

  else if(nu == 0.0) {

    gsl_sf_result K_scaled;

    /* This cannot underflow, and

     * it will not throw GSL_EDOM

     * since that is already checked.

     */

    gsl_sf_bessel_K0_scaled_e(x, &K_scaled);

    result->val  = -x + log(fabs(K_scaled.val));

    result->err  = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);

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

    return GSL_SUCCESS;

  }

  else if(x < 2.0 && nu > 1.0) {

    /* Make use of the inequality

     * Knu(x) <= 1/2 (2/x)^nu Gamma(nu),

     * which follows from the integral representation

     * [Abramowitz+Stegun, 9.6.23 (2)]. With this

     * we decide whether or not there is an overflow

     * problem because x is small.

     */

    double ln_bound;

    gsl_sf_result lg_nu;

    gsl_sf_lngamma_e(nu, &lg_nu);

    ln_bound = -M_LN2 - nu*log(0.5*x) + lg_nu.val;

    if(ln_bound > GSL_LOG_DBL_MAX - 20.0) {

      /* x must be very small or nu very large (or both).

       */

      double xi  = 0.25*x*x;

      double sum = 1.0 - xi/(nu-1.0);

      if(nu > 2.0) sum +=  (xi/(nu-1.0)) * (xi/(nu-2.0));

      result->val  = ln_bound + log(sum);

      result->err  = lg_nu.err;

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

      return GSL_SUCCESS;

    }

    /* can drop-through here */

  }

  {

    /* We passed the above tests, so no problem.

     * Evaluate as usual. Note the possible drop-through

     * in the above code!

     */

    gsl_sf_result_e10 K_scaled;

    int status = gsl_sf_bessel_Knu_scaled_e10_e(nu, x, &K_scaled);

    result->val  = -x + log(fabs(K_scaled.val)) + K_scaled.e10 * M_LN10;

    result->err  = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);

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

    return status;

  }

}

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

#include "eval.h"

double gsl_sf_bessel_Knu_scaled(const double nu, const double x)

{

  EVAL_RESULT(gsl_sf_bessel_Knu_scaled_e(nu, x, &result));

}

double gsl_sf_bessel_Knu(const double nu, const double x)

{

  EVAL_RESULT(gsl_sf_bessel_Knu_e(nu, x, &result));

}

double gsl_sf_bessel_lnKnu(const double nu, const double x)

{

  EVAL_RESULT(gsl_sf_bessel_lnKnu_e(nu, x, &result));

}