/*!
 * \file
 * \brief Implementation of modified Bessel functions of third kind
 * \author Tony Ottosson
 *
 * -------------------------------------------------------------------------
 *
 * IT++ - C++ library of mathematical, signal processing, speech processing,
 *        and communications classes and functions
 *
 * Copyright (C) 1995-2009  (see AUTHORS file for a list of contributors)
 *
 * 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 St, Fifth Floor, Boston, MA 02110-1301 USA
 *
 * -------------------------------------------------------------------------
 *
 * This is slightly modified routine from the Cephes library:
 * http://www.netlib.org/cephes/
 */

#include <itpp/base/bessel/bessel_internal.h>
#include <itpp/base/itassert.h>


/*
 * Modified Bessel function, third kind, integer order
 *
 * double x, y, kn();
 * int n;
 *
 * y = kn( n, x );
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of the third kind
 * of order n of the argument.
 *
 * The range is partitioned into the two intervals [0,9.55] and
 * (9.55, infinity).  An ascending power series is used in the
 * low range, and an asymptotic expansion in the high range.
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        90000       1.8e-8      3.0e-10
 *
 *  Error is high only near the crossover point x = 9.55
 * between the two expansions used.
 */


/*
  Cephes Math Library Release 2.8:  June, 2000
  Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*/


/*
Algorithm for Kn.
                       n-1
                   -n   -  (n-k-1)!    2   k
K (x)  =  0.5 (x/2)     >  -------- (-x /4)
 n                      -     k!
                       k=0

                    inf.                                   2   k
       n         n   -                                   (x /4)
 + (-1)  0.5(x/2)    >  {p(k+1) + p(n+k+1) - 2log(x/2)} ---------
                     -                                  k! (n+k)!
                    k=0

where  p(m) is the psi function: p(1) = -EUL and

                      m-1
                       -
      p(m)  =  -EUL +  >  1/k
                       -
                      k=1

For large x,
                                         2        2     2
                                      u-1     (u-1 )(u-3 )
K (z)  =  sqrt(pi/2z) exp(-z) { 1 + ------- + ------------ + ...}
 v                                        1            2
                                    1! (8z)     2! (8z)
asymptotically, where

           2
    u = 4 v .

*/


#define EUL 5.772156649015328606065e-1
#define MAXFAC 31

#define MACHEP 1.11022302462515654042E-16   /* 2**-53 */
#define MAXLOG 7.08396418532264106224E2     /* log 2**1022 */
#define MINLOG -7.08396418532264106224E2    /* log 2**-1022 */
#define MAXNUM 1.79769313486231570815E308    /* 2**1024*(1-MACHEP) */
#define PI 3.14159265358979323846       /* pi */


double kn(int nn, double x)
{
  double k, kf, nk1f, nkf, zn, t, s, z0, z;
  double ans, fn, pn, pk, zmn, tlg, tox;
  int i, n;

  if (nn < 0)
    n = -nn;
  else
    n = nn;

  if (n > MAXFAC) {
  overf:
    it_warning("kn(): overflow range error");
    return(MAXNUM);
  }

  if (x <= 0.0) {
    if (x < 0.0)
      it_warning("kn(): argument domain error");
    else
      it_warning("kn(): function singularity");
    return(MAXNUM);
  }


  if (x > 9.55)
    goto asymp;

  ans = 0.0;
  z0 = 0.25 * x * x;
  fn = 1.0;
  pn = 0.0;
  zmn = 1.0;
  tox = 2.0 / x;

  if (n > 0) {
    /* compute factorial of n and psi(n) */
    pn = -EUL;
    k = 1.0;
    for (i = 1; i < n; i++) {
      pn += 1.0 / k;
      k += 1.0;
      fn *= k;
    }

    zmn = tox;

    if (n == 1) {
      ans = 1.0 / x;
    }
    else {
      nk1f = fn / n;
      kf = 1.0;
      s = nk1f;
      z = -z0;
      zn = 1.0;
      for (i = 1; i < n; i++) {
        nk1f = nk1f / (n - i);
        kf = kf * i;
        zn *= z;
        t = nk1f * zn / kf;
        s += t;
        if ((MAXNUM - fabs(t)) < fabs(s))
          goto overf;
        if ((tox > 1.0) && ((MAXNUM / tox) < zmn))
          goto overf;
        zmn *= tox;
      }
      s *= 0.5;
      t = fabs(s);
      if ((zmn > 1.0) && ((MAXNUM / zmn) < t))
        goto overf;
      if ((t > 1.0) && ((MAXNUM / t) < zmn))
        goto overf;
      ans = s * zmn;
    }
  }


  tlg = 2.0 * log(0.5 * x);
  pk = -EUL;
  if (n == 0) {
    pn = pk;
    t = 1.0;
  }
  else {
    pn = pn + 1.0 / n;
    t = 1.0 / fn;
  }
  s = (pk + pn - tlg) * t;
  k = 1.0;
  do {
    t *= z0 / (k * (k + n));
    pk += 1.0 / k;
    pn += 1.0 / (k + n);
    s += (pk + pn - tlg) * t;
    k += 1.0;
  }
  while (fabs(t / s) > MACHEP);

  s = 0.5 * s / zmn;
  if (n & 1)
    s = -s;
  ans += s;

  return(ans);



  /* Asymptotic expansion for Kn(x) */
  /* Converges to 1.4e-17 for x > 18.4 */

asymp:

  if (x > MAXLOG) {
    it_warning("kn(): underflow range error");
    return(0.0);
  }
  k = n;
  pn = 4.0 * k * k;
  pk = 1.0;
  z0 = 8.0 * x;
  fn = 1.0;
  t = 1.0;
  s = t;
  nkf = MAXNUM;
  i = 0;
  do {
    z = pn - pk * pk;
    t = t * z / (fn * z0);
    nk1f = fabs(t);
    if ((i >= n) && (nk1f > nkf)) {
      goto adone;
    }
    nkf = nk1f;
    s += t;
    fn += 1.0;
    pk += 2.0;
    i += 1;
  }
  while (fabs(t / s) > MACHEP);

adone:
  ans = exp(-x) * sqrt(PI / (2.0 * x)) * s;
  return(ans);
}
