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/*!
* \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) 1995-2008 (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, order zero
*
* doule x, y, k0();
*
* y = k0( x );
*
* DESCRIPTION:
*
* Returns modified Bessel function of the third kind
* of order zero of the argument.
*
* The range is partitioned into the two intervals [0,8] and
* (8, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
* ACCURACY:
*
* Tested at 2000 random points between 0 and 8. Peak absolute
* error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.2e-15 1.6e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* K0 domain x <= 0 MAXNUM
*/
/*
* Modified Bessel function, third kind, order zero, exponentially scaled
*
* double x, y, k0e();
*
* y = k0e( x );
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of the third kind of order zero of the argument.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.4e-15 1.4e-16
* See k0().
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/
/* Chebyshev coefficients for K0(x) + log(x/2) I0(x)
* in the interval [0,2]. The odd order coefficients are all
* zero; only the even order coefficients are listed.
*
* lim(x->0){ K0(x) + log(x/2) I0(x) } = -EUL.
*/
static double A[] =
{
1.37446543561352307156E-16,
4.25981614279661018399E-14,
1.03496952576338420167E-11,
1.90451637722020886025E-9,
2.53479107902614945675E-7,
2.28621210311945178607E-5,
1.26461541144692592338E-3,
3.59799365153615016266E-2,
3.44289899924628486886E-1,
-5.35327393233902768720E-1
};
/* Chebyshev coefficients for exp(x) sqrt(x) K0(x)
* in the inverted interval [2,infinity].
*
* lim(x->inf){ exp(x) sqrt(x) K0(x) } = sqrt(pi/2).
*/
static double B[] = {
5.30043377268626276149E-18,
-1.64758043015242134646E-17,
5.21039150503902756861E-17,
-1.67823109680541210385E-16,
5.51205597852431940784E-16,
-1.84859337734377901440E-15,
6.34007647740507060557E-15,
-2.22751332699166985548E-14,
8.03289077536357521100E-14,
-2.98009692317273043925E-13,
1.14034058820847496303E-12,
-4.51459788337394416547E-12,
1.85594911495471785253E-11,
-7.95748924447710747776E-11,
3.57739728140030116597E-10,
-1.69753450938905987466E-9,
8.57403401741422608519E-9,
-4.66048989768794782956E-8,
2.76681363944501510342E-7,
-1.83175552271911948767E-6,
1.39498137188764993662E-5,
-1.28495495816278026384E-4,
1.56988388573005337491E-3,
-3.14481013119645005427E-2,
2.44030308206595545468E0
};
#define MAXNUM 1.79769313486231570815E308 /* 2**1024*(1-MACHEP) */
double k0(double x)
{
double y, z;
if( x <= 0.0 )
{
it_warning("besselk:: argument domain error");
//mtherr( "k0", DOMAIN );
return( MAXNUM );
}
if( x <= 2.0 )
{
y = x * x - 2.0;
y = chbevl( y, A, 10 ) - log( 0.5 * x ) * i0(x);
return( y );
}
z = 8.0/x - 2.0;
y = exp(-x) * chbevl( z, B, 25 ) / sqrt(x);
return(y);
}
double k0e(double x)
{
double y;
if( x <= 0.0 )
{
it_warning("besselk:: argument domain error");
//mtherr( "k0e", DOMAIN );
return( MAXNUM );
}
if( x <= 2.0 )
{
y = x * x - 2.0;
y = chbevl( y, A, 10 ) - log( 0.5 * x ) * i0(x);
return( y * exp(x) );
}
y = chbevl( 8.0/x - 2.0, B, 25 ) / sqrt(x);
return(y);
}
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