/* $Id: psi.c,v 1.1 2003/04/15 03:20:17 cher Exp $ */

/* This file is taken from Cephes math library */

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

#include "cephes.h"
#include "mconf.h"
#include <math.h>

/*							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
 */

#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

extern double PI, MAXNUM;

double
cephes_psi(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 ) {
      cephes_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 * cephes_polevl( z, A, 6 );
  } else
    y = 0.0;

  y = log(s)  -  (0.5/s)  -  y  -  w;
  
 done:
  if( negative ) {
    y -= nz;
  }
  return(y);
}

/*
 * Local variables:
 *  compile-command: "make -C .."
 * End:
 */