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/* zetacf.c
*
* Riemann zeta function
*
*
*
* SYNOPSIS:
*
* float x, y, zetacf();
*
* y = zetacf( x );
*
*
*
* DESCRIPTION:
*
*
*
* inf.
* - -x
* zetac(x) = > k , x > 1,
* -
* k=2
*
* is related to the Riemann zeta function by
*
* Riemann zeta(x) = zetac(x) + 1.
*
* Extension of the function definition for x < 1 is implemented.
* Zero is returned for x > log2(MAXNUM).
*
* An overflow error may occur for large negative x, due to the
* gamma function in the reflection formula.
*
* ACCURACY:
*
* Tabulated values have full machine accuracy.
*
* Relative error:
* arithmetic domain # trials peak rms
* IEEE 1,50 30000 5.5e-7 7.5e-8
*
*
*/
/*
Cephes Math Library Release 2.2: July, 1992
Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include <math.h>
/* Riemann zeta(x) - 1
* for integer arguments between 0 and 30.
*/
static float azetacf[] = {
-1.50000000000000000000E0,
1.70141183460469231730E38, /* infinity. */
6.44934066848226436472E-1,
2.02056903159594285400E-1,
8.23232337111381915160E-2,
3.69277551433699263314E-2,
1.73430619844491397145E-2,
8.34927738192282683980E-3,
4.07735619794433937869E-3,
2.00839282608221441785E-3,
9.94575127818085337146E-4,
4.94188604119464558702E-4,
2.46086553308048298638E-4,
1.22713347578489146752E-4,
6.12481350587048292585E-5,
3.05882363070204935517E-5,
1.52822594086518717326E-5,
7.63719763789976227360E-6,
3.81729326499983985646E-6,
1.90821271655393892566E-6,
9.53962033872796113152E-7,
4.76932986787806463117E-7,
2.38450502727732990004E-7,
1.19219925965311073068E-7,
5.96081890512594796124E-8,
2.98035035146522801861E-8,
1.49015548283650412347E-8,
7.45071178983542949198E-9,
3.72533402478845705482E-9,
1.86265972351304900640E-9,
9.31327432419668182872E-10
};
/* 2**x (1 - 1/x) (zeta(x) - 1) = P(1/x)/Q(1/x), 1 <= x <= 10 */
static float P[9] = {
5.85746514569725319540E11,
2.57534127756102572888E11,
4.87781159567948256438E10,
5.15399538023885770696E9,
3.41646073514754094281E8,
1.60837006880656492731E7,
5.92785467342109522998E5,
1.51129169964938823117E4,
2.01822444485997955865E2,
};
static float Q[8] = {
/* 1.00000000000000000000E0,*/
3.90497676373371157516E11,
5.22858235368272161797E10,
5.64451517271280543351E9,
3.39006746015350418834E8,
1.79410371500126453702E7,
5.66666825131384797029E5,
1.60382976810944131506E4,
1.96436237223387314144E2,
};
/* log(zeta(x) - 1 - 2**-x), 10 <= x <= 50 */
static float A[11] = {
8.70728567484590192539E6,
1.76506865670346462757E8,
2.60889506707483264896E10,
5.29806374009894791647E11,
2.26888156119238241487E13,
3.31884402932705083599E14,
5.13778997975868230192E15,
-1.98123688133907171455E15,
-9.92763810039983572356E16,
7.82905376180870586444E16,
9.26786275768927717187E16,
};
static float B[10] = {
/* 1.00000000000000000000E0,*/
-7.92625410563741062861E6,
-1.60529969932920229676E8,
-2.37669260975543221788E10,
-4.80319584350455169857E11,
-2.07820961754173320170E13,
-2.96075404507272223680E14,
-4.86299103694609136686E15,
5.34589509675789930199E15,
5.71464111092297631292E16,
-1.79915597658676556828E16,
};
/* (1-x) (zeta(x) - 1), 0 <= x <= 1 */
static float R[6] = {
-3.28717474506562731748E-1,
1.55162528742623950834E1,
-2.48762831680821954401E2,
1.01050368053237678329E3,
1.26726061410235149405E4,
-1.11578094770515181334E5,
};
static float S[5] = {
/* 1.00000000000000000000E0,*/
1.95107674914060531512E1,
3.17710311750646984099E2,
3.03835500874445748734E3,
2.03665876435770579345E4,
7.43853965136767874343E4,
};
#define MAXL2 127
/*
* Riemann zeta function, minus one
*/
extern float MACHEPF, PIO2F, MAXNUMF, PIF;
#ifdef ANSIC
extern float sinf ( float xx );
extern float floorf ( float x );
extern float gammaf ( float xx );
extern float powf ( float x, float y );
extern float expf ( float xx );
extern float polevlf ( float xx, float *coef, int N );
extern float p1evlf ( float xx, float *coef, int N );
#else
float sinf(), floorf(), gammaf(), powf(), expf();
float polevlf(), p1evlf();
#endif
float zetacf(float xx)
{
int i;
float x, a, b, s, w;
x = xx;
if( x < 0.0 )
{
if( x < -30.8148 )
{
mtherr( "zetacf", OVERFLOW );
return(0.0);
}
s = 1.0 - x;
w = zetacf( s );
b = sinf(PIO2F*x) * powf(2.0*PIF, x) * gammaf(s) * (1.0 + w) / PIF;
return(b - 1.0);
}
if( x >= MAXL2 )
return(0.0); /* because first term is 2**-x */
/* Tabulated values for integer argument */
w = floorf(x);
if( w == x )
{
i = x;
if( i < 31 )
{
return( azetacf[i] );
}
}
if( x < 1.0 )
{
w = 1.0 - x;
a = polevlf( x, R, 5 ) / ( w * p1evlf( x, S, 5 ));
return( a );
}
if( x == 1.0 )
{
mtherr( "zetacf", SING );
return( MAXNUMF );
}
if( x <= 10.0 )
{
b = powf( 2.0, x ) * (x - 1.0);
w = 1.0/x;
s = (x * polevlf( w, P, 8 )) / (b * p1evlf( w, Q, 8 ));
return( s );
}
if( x <= 50.0 )
{
b = powf( 2.0, -x );
w = polevlf( x, A, 10 ) / p1evlf( x, B, 10 );
w = expf(w) + b;
return(w);
}
/* Basic sum of inverse powers */
s = 0.0;
a = 1.0;
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);
}
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