uclibc-ng/libm/float/zetaf.c

/*							zetaf.c
 *
 *	Riemann zeta function of two arguments
 *
 *
 *
 * SYNOPSIS:
 *
 * float x, q, y, zetaf();
 *
 * y = zetaf( 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:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,25        10000       6.9e-7      1.0e-7
 *
 * Large arguments may produce underflow in powf(), in which
 * case the results are inaccurate.
 *
 * REFERENCE:
 *
 * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
 * Series, and Products, p. 1073; Academic Press, 1980.
 *
 */

/*
Cephes Math Library Release 2.2:  July, 1992
Copyright 1984, 1987, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include <math.h>
extern float MAXNUMF, MACHEPF;

/* Expansion coefficients
 * for Euler-Maclaurin summation formula
 * (2k)! / B2k
 * where B2k are Bernoulli numbers
 */
static float 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 */


#define fabsf(x) ( (x) < 0 ? -(x) : (x) )


float powf( float, float );
float zetaf(float xx, float qq)
{
int i;
float x, q, a, b, k, s, w, t;

x = xx;
q = qq;
if( x == 1.0 )
	return( MAXNUMF );

if( x < 1.0 )
	{
	mtherr( "zetaf", DOMAIN );
	return(0.0);
	}


/* Euler-Maclaurin summation formula */
/*
if( x < 25.0 )
{
*/
w = 9.0;
s = powf( q, -x );
a = q;
for( i=0; i<9; i++ )
	{
	a += 1.0;
	b = powf( a, -x );
	s += b;
	if( b/s < MACHEPF )
		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 = fabsf(t/s);
	if( t < MACHEPF )
		goto done;
	k += 1.0;
	a *= x + k;
	b /= w;
	k += 1.0;
	}
done:
return(s);
/*
}
*/


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

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

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