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/* k0.c
*
* Modified Bessel function, third kind, order zero
*
*
*
* SYNOPSIS:
*
* double 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 NPY_INFINITY
*
*/
/* k0e()
*
* Modified Bessel function, third kind, order zero,
* exponentially scaled
*
*
*
* SYNOPSIS:
*
* 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
*/
#include "mconf.h"
/* 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
};
double k0(x)
double x;
{
double y, z;
if (x == 0.0) {
mtherr("k0", SING);
return NPY_INFINITY;
}
else if (x < 0.0) {
mtherr("k0", DOMAIN);
return NPY_NAN;
}
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(x)
double x;
{
double y;
if (x == 0.0) {
mtherr("k0e", SING);
return NPY_INFINITY;
}
else if (x < 0.0) {
mtherr("k0e", DOMAIN);
return NPY_NAN;
}
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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