iv.cpp

强大的C++库

/*!
 * \file 
 * \brief Implementation of modified Bessel functions of noninteager order
 * \author Tony Ottosson
 *
 * $Date: 2006-04-03 15:34:15 +0200 (Mon, 03 Apr 2006) $
 * $Revision: 400 $
 *
 * -------------------------------------------------------------------------
 *
 * IT++ - C++ library of mathematical, signal processing, speech processing,
 *        and communications classes and functions
 *
 * Copyright (C) 1995-2005  (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/scalfunc.h>
#include <cmath>

using namespace itpp;

/* 
 * Modified Bessel function of noninteger order
 *
 * double v, x, y, iv();
 *
 * y = iv( v, x );
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order v of the
 * argument.  If x is negative, v must be integer valued.
 *
 * The function is defined as Iv(x) = Jv( ix ).  It is
 * here computed in terms of the confluent hypergeometric
 * function, according to the formula
 *
 *              v  -x
 * Iv(x) = (x/2)  e   hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
 *
 * If v is a negative integer, then v is replaced by -v.
 *
 *
 * ACCURACY:
 *
 * Tested at random points (v, x), with v between 0 and
 * 30, x between 0 and 28.
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30         10000      1.7e-14     2.7e-15
 *
 * Accuracy is diminished if v is near a negative integer.
 *
 * See also hyperg.c.
 */

/* Mdified Bessel function of noninteger order */
/* If x < 0, then v must be an integer. */

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


#define MAXNUM 1.79769313486231570815E308    /* 2**1024*(1-MACHEP) */

double iv(double v, double x)
{
  int sign;
  double t, ax;

  /* If v is a negative integer, invoke symmetry */
  t = floor(v);
  if( v < 0.0 )
    {
      if( t == v )
	{
	  v = -v;	/* symmetry */
	  t = -t;
	}
    }
  /* If x is negative, require v to be an integer */
  sign = 1;
  if( x < 0.0 )
    {
      if( t != v )
	{
	  it_warning("besseli:: argument domain error");
	  //mtherr( "iv", DOMAIN );
	  return( 0.0 );
	}
      if( v != 2.0 * floor(v/2.0) )
	sign = -1;
    }

  /* Avoid logarithm singularity */
  if( x == 0.0 )
    {
      if( v == 0.0 )
	return( 1.0 );
      if( v < 0.0 )
	{
	  it_warning("besseli:: overflow");
	  //mtherr( "iv", OVERFLOW );
	  return( MAXNUM );
	}
      else
	return( 0.0 );
    }

  ax = fabs(x);
  t = v * log( 0.5 * ax )  -  x;
  t = sign * exp(t) / itpp::gamma( v + 1.0 );
  ax = v + 0.5;
  return( t * hyperg( ax,  2.0 * ax,  2.0 * x ) );
}