source: sasview/sansmodels/src/cephes/kn.c @ 476977b

/*                                                      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
 *    DEC       0,30         3000       1.3e-9      5.8e-11
 *    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"

#define EUL 5.772156649015328606065e-1
#define MAXFAC 31
#ifdef ANSIPROT
extern double fabs ( double );
extern double exp ( double );
extern double log ( double );
extern double sqrt ( double );
#else
double fabs(), exp(), log(), sqrt();
#endif
extern double MACHEP, MAXNUM, MAXLOG, PI;

double kn( nn, x )
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:
        mtherr( "kn", OVERFLOW );
        return( MAXNUM );
        }

if( x <= 0.0 )
        {
        if( x < 0.0 )
                mtherr( "kn", DOMAIN );
        else
                mtherr( "kn", SING );
        return( MAXNUM );
        }


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( (MAXNUM - fabs(t)) < fabs(s) )
                                goto overf;
                        if( (tox > 1.0) && ((MAXNUM/tox) < zmn) )
                                goto overf;
                        zmn *= tox;
                        }
                s *= 0.5;
                t = fabs(s);
                if( (zmn > 1.0) && ((MAXNUM/zmn) < t) )
                        goto overf;
                if( (t > 1.0) && ((MAXNUM/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 )
        {
        mtherr( "kn", UNDERFLOW );
        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 = MAXNUM;
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( PI/(2.0*x) ) * s;
return(ans);
}