root/library/bdm/base/itpp/base/bessel/i0.cpp @ 813

/*!
 * \file
 * \brief Implementation of modified Bessel functions of order zero
 * \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>


/*
 *
 * Modified Bessel function of order zero
 *
 * double x, y, i0();
 *
 * y = i0( x );
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order zero of the
 * argument.
 *
 * The function is defined as i0(x) = j0( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity).  Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        30000       5.8e-16     1.4e-16
 *
 */

/*
 * Modified Bessel function of order zero,
 * exponentially scaled
 *
 * double x, y, i0e();
 *
 * y = i0e( x );
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of order zero of the argument.
 *
 * The function is defined as i0e(x) = exp(-|x|) j0( ix ).
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        30000       5.4e-16     1.2e-16
 * See i0().
 */

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


/* Chebyshev coefficients for exp(-x) I0(x)
 * in the interval [0,8].
 *
 * lim(x->0){ exp(-x) I0(x) } = 1.
 */

static double A[] = {
  -4.41534164647933937950E-18,
  3.33079451882223809783E-17,
  -2.43127984654795469359E-16,
  1.71539128555513303061E-15,
  -1.16853328779934516808E-14,
  7.67618549860493561688E-14,
  -4.85644678311192946090E-13,
  2.95505266312963983461E-12,
  -1.72682629144155570723E-11,
  9.67580903537323691224E-11,
  -5.18979560163526290666E-10,
  2.65982372468238665035E-9,
  -1.30002500998624804212E-8,
  6.04699502254191894932E-8,
  -2.67079385394061173391E-7,
  1.11738753912010371815E-6,
  -4.41673835845875056359E-6,
  1.64484480707288970893E-5,
  -5.75419501008210370398E-5,
  1.88502885095841655729E-4,
  -5.76375574538582365885E-4,
  1.63947561694133579842E-3,
  -4.32430999505057594430E-3,
  1.05464603945949983183E-2,
  -2.37374148058994688156E-2,
  4.93052842396707084878E-2,
  -9.49010970480476444210E-2,
  1.71620901522208775349E-1,
  -3.04682672343198398683E-1,
  6.76795274409476084995E-1
};



/* Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
 * in the inverted interval [8,infinity].
 *
 * lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
 */

static double B[] = {
  -7.23318048787475395456E-18,
  -4.83050448594418207126E-18,
  4.46562142029675999901E-17,
  3.46122286769746109310E-17,
  -2.82762398051658348494E-16,
  -3.42548561967721913462E-16,
  1.77256013305652638360E-15,
  3.81168066935262242075E-15,
  -9.55484669882830764870E-15,
  -4.15056934728722208663E-14,
  1.54008621752140982691E-14,
  3.85277838274214270114E-13,
  7.18012445138366623367E-13,
  -1.79417853150680611778E-12,
  -1.32158118404477131188E-11,
  -3.14991652796324136454E-11,
  1.18891471078464383424E-11,
  4.94060238822496958910E-10,
  3.39623202570838634515E-9,
  2.26666899049817806459E-8,
  2.04891858946906374183E-7,
  2.89137052083475648297E-6,
  6.88975834691682398426E-5,
  3.36911647825569408990E-3,
  8.04490411014108831608E-1
};


double i0(double x)
{
  double y;

  if (x < 0)
    x = -x;
  if (x <= 8.0) {
    y = (x / 2.0) - 2.0;
    return(exp(x) * chbevl(y, A, 30));
  }

  return(exp(x) * chbevl(32.0 / x - 2.0, B, 25) / sqrt(x));

}


double i0e(double x)
{
  double y;

  if (x < 0)
    x = -x;
  if (x <= 8.0) {
    y = (x / 2.0) - 2.0;
    return(chbevl(y, A, 30));
  }

  return(chbevl(32.0 / x - 2.0, B, 25) / sqrt(x));

}