k0.c

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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          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(double x)
{
    double y, z;
 
    if (x == 0.0) {
        sf_error("k0", SF_ERROR_SINGULAR, NULL);
        return INFINITY;
    }
    else if (x < 0.0) {
        sf_error("k0", SF_ERROR_DOMAIN, NULL);
        return 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(double x)
{
    double y;
 
    if (x == 0.0) {
        sf_error("k0e", SF_ERROR_SINGULAR, NULL);
        return INFINITY;
    }
    else if (x < 0.0) {
        sf_error("k0e", SF_ERROR_DOMAIN, NULL);
        return 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);
}