kn.c

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/*                                                     kn.c
 *
 *     Modified Bessel function, third kind, integer order
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, kn();
 * int n;
 *
 * y = kn( n, x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns modified Bessel function of the third kind
 * of order n of the argument.
 *
 * The range is partitioned into the two intervals [0,9.55] and
 * (9.55, infinity).  An ascending power series is used in the
 * low range, and an asymptotic expansion in the high range.
 *
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30        90000       1.8e-8      3.0e-10
 *
 *  Error is high only near the crossover point x = 9.55
 * between the two expansions used.
 */

 
/*
 * Cephes Math Library Release 2.8:  June, 2000
 * Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
 */
 
 
/*
 * Algorithm for Kn.
 *                        n-1
 *                    -n   -  (n-k-1)!    2   k
 * K (x)  =  0.5 (x/2)     >  -------- (-x /4)
 *  n                      -     k!
 *                        k=0
 *
 *                     inf.                                   2   k
 *        n         n   -                                   (x /4)
 *  + (-1)  0.5(x/2)    >  {p(k+1) + p(n+k+1) - 2log(x/2)} ---------
 *                      -                                  k! (n+k)!
 *                     k=0
 *
 * where  p(m) is the psi function: p(1) = -EUL and
 *
 *                       m-1
 *                        -
 *       p(m)  =  -EUL +  >  1/k
 *                        -
 *                       k=1
 *
 * For large x,
 *                                          2        2     2
 *                                       u-1     (u-1 )(u-3 )
 * K (z)  =  sqrt(pi/2z) exp(-z) { 1 + ------- + ------------ + ...}
 *  v                                        1            2
 *                                     1! (8z)     2! (8z)
 * asymptotically, where
 *
 *            2
 *     u = 4 v .
 *
 */
 
#include "mconf.h"
#include <float.h>
 
#define EUL 5.772156649015328606065e-1
#define MAXFAC 31
extern double MACHEP, MAXLOG;
 
double kn(int nn, double x)
{
    double k, kf, nk1f, nkf, zn, t, s, z0, z;
    double ans, fn, pn, pk, zmn, tlg, tox;
    int i, n;
 
    if (nn < 0)
        n = -nn;
    else
        n = nn;
 
    if (n > MAXFAC) {
      overf:
        sf_error("kn", SF_ERROR_OVERFLOW, NULL);
        return (INFINITY);
    }
 
    if (x <= 0.0) {
        if (x < 0.0) {
            sf_error("kn", SF_ERROR_DOMAIN, NULL);
            return NAN;
        }
        else {
            sf_error("kn", SF_ERROR_SINGULAR, NULL);
            return INFINITY;
        }
    }
 
 
    if (x > 9.55)
        goto asymp;
 
    ans = 0.0;
    z0 = 0.25 * x * x;
    fn = 1.0;
    pn = 0.0;
    zmn = 1.0;
    tox = 2.0 / x;
 
    if (n > 0) {
        /* compute factorial of n and psi(n) */
        pn = -EUL;
        k = 1.0;
        for (i = 1; i < n; i++) {
            pn += 1.0 / k;
            k += 1.0;
            fn *= k;
        }
 
        zmn = tox;
 
        if (n == 1) {
            ans = 1.0 / x;
        }
        else {
            nk1f = fn / n;
            kf = 1.0;
            s = nk1f;
            z = -z0;
            zn = 1.0;
            for (i = 1; i < n; i++) {
                nk1f = nk1f / (n - i);
                kf = kf * i;
                zn *= z;
                t = nk1f * zn / kf;
                s += t;
                if ((DBL_MAX - fabs(t)) < fabs(s))
                    goto overf;
                if ((tox > 1.0) && ((DBL_MAX / tox) < zmn))
                    goto overf;
                zmn *= tox;
            }
            s *= 0.5;
            t = fabs(s);
            if ((zmn > 1.0) && ((DBL_MAX / zmn) < t))
                goto overf;
            if ((t > 1.0) && ((DBL_MAX / t) < zmn))
                goto overf;
            ans = s * zmn;
        }
    }
 
 
    tlg = 2.0 * log(0.5 * x);
    pk = -EUL;
    if (n == 0) {
        pn = pk;
        t = 1.0;
    }
    else {
        pn = pn + 1.0 / n;
        t = 1.0 / fn;
    }
    s = (pk + pn - tlg) * t;
    k = 1.0;
    do {
        t *= z0 / (k * (k + n));
        pk += 1.0 / k;
        pn += 1.0 / (k + n);
        s += (pk + pn - tlg) * t;
        k += 1.0;
    }
    while (fabs(t / s) > MACHEP);
 
    s = 0.5 * s / zmn;
    if (n & 1)
        s = -s;
    ans += s;
 
    return (ans);
 
 
 
    /* Asymptotic expansion for Kn(x) */
    /* Converges to 1.4e-17 for x > 18.4 */
 
  asymp:
 
    if (x > MAXLOG) {
        sf_error("kn", SF_ERROR_UNDERFLOW, NULL);
        return (0.0);
    }
    k = n;
    pn = 4.0 * k * k;
    pk = 1.0;
    z0 = 8.0 * x;
    fn = 1.0;
    t = 1.0;
    s = t;
    nkf = INFINITY;
    i = 0;
    do {
        z = pn - pk * pk;
        t = t * z / (fn * z0);
        nk1f = fabs(t);
        if ((i >= n) && (nk1f > nkf)) {
            goto adone;
        }
        nkf = nk1f;
        s += t;
        fn += 1.0;
        pk += 2.0;
        i += 1;
    }
    while (fabs(t / s) > MACHEP);
 
  adone:
    ans = exp(-x) * sqrt(M_PI / (2.0 * x)) * s;
    return (ans);
}