CocoRoboDesktop/pdf1.js/node_modules/cephes/cephes/psi.c

/*							psi.c
 *
 *	Psi (digamma) function
 *
 *
 * SYNOPSIS:
 *
 * double x, y, psi();
 *
 * y = psi( x );
 *
 *
 * DESCRIPTION:
 *
 *              d      -
 *   psi(x)  =  -- ln | (x)
 *              dx
 *
 * is the logarithmic derivative of the gamma function.
 * For integer x,
 *                   n-1
 *                    -
 * psi(n) = -EUL  +   >  1/k.
 *                    -
 *                   k=1
 *
 * This formula is used for 0 < n <= 10.  If x is negative, it
 * is transformed to a positive argument by the reflection
 * formula  psi(1-x) = psi(x) + pi cot(pi x).
 * For general positive x, the argument is made greater than 10
 * using the recurrence  psi(x+1) = psi(x) + 1/x.
 * Then the following asymptotic expansion is applied:
 *
 *                           inf.   B
 *                            -      2k
 * psi(x) = log(x) - 1/2x -   >   -------
 *                            -        2k
 *                           k=1   2k x
 *
 * where the B2k are Bernoulli numbers.
 *
 * ACCURACY:
 *    Relative error (except absolute when |psi| < 1):
 * arithmetic   domain     # trials      peak         rms
 *    DEC       0,30         2500       1.7e-16     2.0e-17
 *    IEEE      0,30        30000       1.3e-15     1.4e-16
 *    IEEE      -30,0       40000       1.5e-15     2.2e-16
 *
 * ERROR MESSAGES:
 *     message         condition      value returned
 * psi singularity    x integer <=0      MAXNUM
 */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

#ifdef UNK
static double A[] = {8.33333333333333333333E-2, -2.10927960927960927961E-2,
                     7.57575757575757575758E-3, -4.16666666666666666667E-3,
                     3.96825396825396825397E-3, -8.33333333333333333333E-3,
                     8.33333333333333333333E-2};
#endif

#ifdef DEC
static unsigned short A[] = {
    0037252, 0125252, 0125252, 0125253, 0136654, 0145314, 0126312,
    0146255, 0036370, 0037017, 0101740, 0174076, 0136210, 0104210,
    0104210, 0104211, 0036202, 0004040, 0101010, 0020202, 0136410,
    0104210, 0104210, 0104211, 0037252, 0125252, 0125252, 0125253};
#endif

#ifdef IBMPC
static unsigned short A[] = {0x5555, 0x5555, 0x5555, 0x3fb5, 0x5996, 0x9599,
                             0x9959, 0xbf95, 0x1f08, 0xf07c, 0x07c1, 0x3f7f,
                             0x1111, 0x1111, 0x1111, 0xbf71, 0x0410, 0x1041,
                             0x4104, 0x3f70, 0x1111, 0x1111, 0x1111, 0xbf81,
                             0x5555, 0x5555, 0x5555, 0x3fb5};
#endif

#ifdef MIEEE
static unsigned short A[] = {0x3fb5, 0x5555, 0x5555, 0x5555, 0xbf95, 0x9959,
                             0x9599, 0x5996, 0x3f7f, 0x07c1, 0xf07c, 0x1f08,
                             0xbf71, 0x1111, 0x1111, 0x1111, 0x3f70, 0x4104,
                             0x1041, 0x0410, 0xbf81, 0x1111, 0x1111, 0x1111,
                             0x3fb5, 0x5555, 0x5555, 0x5555};
#endif

#define EUL 0.57721566490153286061

#ifdef ANSIPROT
extern double floor(double);
extern double log(double);
extern double tan(double);
extern double polevl(double, void *, int);
#else
double floor(), log(), tan(), polevl();
#endif
extern double PI, MAXNUM;

double psi(x) double x;
{
  double p, q, nz, s, w, y, z;
  int i, n, negative;

  negative = 0;
  nz = 0.0;

  if (x <= 0.0) {
    negative = 1;
    q = x;
    p = floor(q);
    if (p == q) {
      mtherr("psi", SING);
      return (MAXNUM);
    }
    /* Remove the zeros of tan(PI x)
     * by subtracting the nearest integer from x
     */
    nz = q - p;
    if (nz != 0.5) {
      if (nz > 0.5) {
        p += 1.0;
        nz = q - p;
      }
      nz = PI / tan(PI * nz);
    } else {
      nz = 0.0;
    }
    x = 1.0 - x;
  }

  /* check for positive integer up to 10 */
  if ((x <= 10.0) && (x == floor(x))) {
    y = 0.0;
    n = x;
    for (i = 1; i < n; i++) {
      w = i;
      y += 1.0 / w;
    }
    y -= EUL;
    goto done;
  }

  s = x;
  w = 0.0;
  while (s < 10.0) {
    w += 1.0 / s;
    s += 1.0;
  }

  if (s < 1.0e17) {
    z = 1.0 / (s * s);
    y = z * polevl(z, A, 6);
  } else
    y = 0.0;

  y = log(s) - (0.5 / s) - y - w;

done:

  if (negative) {
    y -= nz;
  }

  return (y);
}