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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"

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

#define EUL 0.57721566490153286061

double psi (double x)
{
    double p, q, nz, s, w, y, z;
    int i, n, negative = 0;

    nz = 0.0;

    if (x <= 0.0) {
      negative = 1;
      q = x;
      p = floor(q);
      if (p == q) {
          mtherr("psi", CEPHES_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;
}

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