uclibc-ng/libm/double/zeta.c

/*							zeta.c
 *
 *	Riemann zeta function of two arguments
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, q, y, zeta();
 *
 * y = zeta( x, q );
 *
 *
 *
 * DESCRIPTION:
 *
 *
 *
 *                 inf.
 *                  -        -x
 *   zeta(x,q)  =   >   (k+q)  
 *                  -
 *                 k=0
 *
 * where x > 1 and q is not a negative integer or zero.
 * The Euler-Maclaurin summation formula is used to obtain
 * the expansion
 *
 *                n         
 *                -       -x
 * zeta(x,q)  =   >  (k+q)  
 *                -         
 *               k=1        
 *
 *           1-x                 inf.  B   x(x+1)...(x+2j)
 *      (n+q)           1         -     2j
 *  +  ---------  -  -------  +   >    --------------------
 *        x-1              x      -                   x+2j+1
 *                   2(n+q)      j=1       (2j)! (n+q)
 *
 * where the B2j are Bernoulli numbers.  Note that (see zetac.c)
 * zeta(x,1) = zetac(x) + 1.
 *
 *
 *
 * ACCURACY:
 *
 *
 *
 * REFERENCE:
 *
 * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
 * Series, and Products, p. 1073; Academic Press, 1980.
 *
 */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*/

#include <math.h>
#ifdef ANSIPROT
extern double fabs ( double );
extern double pow ( double, double );
extern double floor ( double );
#else
double fabs(), pow(), floor();
#endif
extern double MAXNUM, MACHEP;

/* Expansion coefficients
 * for Euler-Maclaurin summation formula
 * (2k)! / B2k
 * where B2k are Bernoulli numbers
 */
static double A[] = {
12.0,
-720.0,
30240.0,
-1209600.0,
47900160.0,
-1.8924375803183791606e9, /*1.307674368e12/691*/
7.47242496e10,
-2.950130727918164224e12, /*1.067062284288e16/3617*/
1.1646782814350067249e14, /*5.109094217170944e18/43867*/
-4.5979787224074726105e15, /*8.028576626982912e20/174611*/
1.8152105401943546773e17, /*1.5511210043330985984e23/854513*/
-7.1661652561756670113e18 /*1.6938241367317436694528e27/236364091*/
};
/* 30 Nov 86 -- error in third coefficient fixed */


double zeta(x,q)
double x,q;
{
int i;
double a, b, k, s, t, w;

if( x == 1.0 )
	goto retinf;

if( x < 1.0 )
	{
domerr:
	mtherr( "zeta", DOMAIN );
	return(0.0);
	}

if( q <= 0.0 )
	{
	if(q == floor(q))
		{
		mtherr( "zeta", SING );
retinf:
		return( MAXNUM );
		}
	if( x != floor(x) )
		goto domerr; /* because q^-x not defined */
	}

/* Euler-Maclaurin summation formula */
/*
if( x < 25.0 )
*/
{
/* Permit negative q but continue sum until n+q > +9 .
 * This case should be handled by a reflection formula.
 * If q<0 and x is an integer, there is a relation to
 * the polygamma function.
 */
s = pow( q, -x );
a = q;
i = 0;
b = 0.0;
while( (i < 9) || (a <= 9.0) )
	{
	i += 1;
	a += 1.0;
	b = pow( a, -x );
	s += b;
	if( fabs(b/s) < MACHEP )
		goto done;
	}

w = a;
s += b*w/(x-1.0);
s -= 0.5 * b;
a = 1.0;
k = 0.0;
for( i=0; i<12; i++ )
	{
	a *= x + k;
	b /= w;
	t = a*b/A[i];
	s = s + t;
	t = fabs(t/s);
	if( t < MACHEP )
		goto done;
	k += 1.0;
	a *= x + k;
	b /= w;
	k += 1.0;
	}
done:
return(s);
}



/* Basic sum of inverse powers */
/*
pseres:

s = pow( q, -x );
a = q;
do
	{
	a += 2.0;
	b = pow( a, -x );
	s += b;
	}
while( b/s > MACHEP );

b = pow( 2.0, -x );
s = (s + b)/(1.0-b);
return(s);
*/
}