i0.c

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/*                                                     i0.c
 *
 *     Modified Bessel function of order zero
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, i0();
 *
 * y = i0( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of order zero of the
 * argument.
 *
 * The function is defined as i0(x) = j0( ix ).
 *
 * The range is partitioned into the two intervals [0,8] and
 * (8, infinity).  Chebyshev polynomial expansions are employed
 * in each interval.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        30000       5.8e-16     1.4e-16
 *
 */
/*                                                     i0e.c
 *
 *      Modified Bessel function of order zero,
 *      exponentially scaled
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, i0e();
 *
 * y = i0e( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns exponentially scaled modified Bessel function
 * of order zero of the argument.
 *
 * The function is defined as i0e(x) = exp(-|x|) j0( ix ).
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        30000       5.4e-16     1.2e-16
 * See i0().
 *
 */

/*                                                     i0.c            */
 
 
/*
 * Cephes Math Library Release 2.8:  June, 2000
 * Copyright 1984, 1987, 2000 by Stephen L. Moshier
 */
 
#include "mconf.h"
 
/* Chebyshev coefficients for exp(-x) I0(x)
 * in the interval [0,8].
 *
 * lim(x->0){ exp(-x) I0(x) } = 1.
 */
static double A[] = {
    -4.41534164647933937950E-18,
    3.33079451882223809783E-17,
    -2.43127984654795469359E-16,
    1.71539128555513303061E-15,
    -1.16853328779934516808E-14,
    7.67618549860493561688E-14,
    -4.85644678311192946090E-13,
    2.95505266312963983461E-12,
    -1.72682629144155570723E-11,
    9.67580903537323691224E-11,
    -5.18979560163526290666E-10,
    2.65982372468238665035E-9,
    -1.30002500998624804212E-8,
    6.04699502254191894932E-8,
    -2.67079385394061173391E-7,
    1.11738753912010371815E-6,
    -4.41673835845875056359E-6,
    1.64484480707288970893E-5,
    -5.75419501008210370398E-5,
    1.88502885095841655729E-4,
    -5.76375574538582365885E-4,
    1.63947561694133579842E-3,
    -4.32430999505057594430E-3,
    1.05464603945949983183E-2,
    -2.37374148058994688156E-2,
    4.93052842396707084878E-2,
    -9.49010970480476444210E-2,
    1.71620901522208775349E-1,
    -3.04682672343198398683E-1,
    6.76795274409476084995E-1
};
 
/* Chebyshev coefficients for exp(-x) sqrt(x) I0(x)
 * in the inverted interval [8,infinity].
 *
 * lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi).
 */
static double B[] = {
    -7.23318048787475395456E-18,
    -4.83050448594418207126E-18,
    4.46562142029675999901E-17,
    3.46122286769746109310E-17,
    -2.82762398051658348494E-16,
    -3.42548561967721913462E-16,
    1.77256013305652638360E-15,
    3.81168066935262242075E-15,
    -9.55484669882830764870E-15,
    -4.15056934728722208663E-14,
    1.54008621752140982691E-14,
    3.85277838274214270114E-13,
    7.18012445138366623367E-13,
    -1.79417853150680611778E-12,
    -1.32158118404477131188E-11,
    -3.14991652796324136454E-11,
    1.18891471078464383424E-11,
    4.94060238822496958910E-10,
    3.39623202570838634515E-9,
    2.26666899049817806459E-8,
    2.04891858946906374183E-7,
    2.89137052083475648297E-6,
    6.88975834691682398426E-5,
    3.36911647825569408990E-3,
    8.04490411014108831608E-1
};
 
double i0(double x)
{
    double y;
 
    if (x < 0)
        x = -x;
    if (x <= 8.0) {
        y = (x / 2.0) - 2.0;
        return (exp(x) * chbevl(y, A, 30));
    }
 
    return (exp(x) * chbevl(32.0 / x - 2.0, B, 25) / sqrt(x));
 
}
 
 
 
 
double i0e(double x)
{
    double y;
 
    if (x < 0)
        x = -x;
    if (x <= 8.0) {
        y = (x / 2.0) - 2.0;
        return (chbevl(y, A, 30));
    }
 
    return (chbevl(32.0 / x - 2.0, B, 25) / sqrt(x));
 
}