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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 one (should be second kind?)
*
* double x, y, k1();
*
* y = k1( x );
*
* DESCRIPTION:
*
* Computes the modified Bessel function of the third kind
* of order one of the argument.
*
* The range is partitioned into the two intervals [0,2] and
* (2, infinity). Chebyshev polynomial expansions are employed
* in each interval.
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 1.2e-15 1.6e-16
*
* ERROR MESSAGES:
*
* message condition value returned
* k1 domain x <= 0 MAXNUM
*/
/*
* Modified Bessel function, third kind, order one, exponentially scaled
*
* SYNOPSIS:
*
* double x, y, k1e();
*
* y = k1e( x );
*
* DESCRIPTION:
*
* Returns exponentially scaled modified Bessel function
* of the third kind of order one of the argument:
*
* k1e(x) = exp(x) * k1(x).
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 0, 30 30000 7.8e-16 1.2e-16
* See k1().
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/
/* Chebyshev coefficients for x(K1(x) - log(x/2) I1(x))
* in the interval [0,2].
*
* lim(x->0){ x(K1(x) - log(x/2) I1(x)) } = 1.
*/
static double A[] =
{
-7.02386347938628759343E-18,
-2.42744985051936593393E-15,
-6.66690169419932900609E-13,
-1.41148839263352776110E-10,
-2.21338763073472585583E-8,
-2.43340614156596823496E-6,
-1.73028895751305206302E-4,
-6.97572385963986435018E-3,
-1.22611180822657148235E-1,
-3.53155960776544875667E-1,
1.52530022733894777053E0
};
/* Chebyshev coefficients for exp(x) sqrt(x) K1(x)
* in the interval [2,infinity].
*
* lim(x->inf){ exp(x) sqrt(x) K1(x) } = sqrt(pi/2).
*/
static double B[] =
{
-5.75674448366501715755E-18,
1.79405087314755922667E-17,
-5.68946255844285935196E-17,
1.83809354436663880070E-16,
-6.05704724837331885336E-16,
2.03870316562433424052E-15,
-7.01983709041831346144E-15,
2.47715442448130437068E-14,
-8.97670518232499435011E-14,
3.34841966607842919884E-13,
-1.28917396095102890680E-12,
5.13963967348173025100E-12,
-2.12996783842756842877E-11,
9.21831518760500529508E-11,
-4.19035475934189648750E-10,
2.01504975519703286596E-9,
-1.03457624656780970260E-8,
5.74108412545004946722E-8,
-3.50196060308781257119E-7,
2.40648494783721712015E-6,
-1.93619797416608296024E-5,
1.95215518471351631108E-4,
-2.85781685962277938680E-3,
1.03923736576817238437E-1,
2.72062619048444266945E0
};
#define MAXNUM 1.79769313486231570815E308 /* 2**1024*(1-MACHEP) */
double k1(double x)
{
double y, z;
z = 0.5 * x;
if( z <= 0.0 )
{
it_warning("k1(): argument domain error");
return( MAXNUM );
}
if( x <= 2.0 )
{
y = x * x - 2.0;
y = log(z) * i1(x) + chbevl( y, A, 11 ) / x;
return( y );
}
return( exp(-x) * chbevl( 8.0/x - 2.0, B, 25 ) / sqrt(x) );
}
double k1e(double x)
{
double y;
if( x <= 0.0 )
{
it_warning("k1e(): argument domain error");
return( MAXNUM );
}
if( x <= 2.0 )
{
y = x * x - 2.0;
y = log( 0.5 * x ) * i1(x) + chbevl( y, A, 11 ) / x;
return( y * exp(x) );
}
return( chbevl( 8.0/x - 2.0, B, 25 ) / sqrt(x) );
}
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