CocoRoboDesktop/pdf1.js/node_modules/cephes/cephes/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 "mconf.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);
  */
}