/*							igam.c
 *
 *	Incomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igam();
 *
 * y = igam( a, x );
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *                           x
 *                            -
 *                   1       | |  -t  a-1
 *  igam(a,x)  =   -----     |   e   t   dt.
 *                  -      | |
 *                 | (a)    -
 *                           0
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30       200000       3.6e-14     2.9e-15
 *    IEEE      0,100      300000       9.9e-14     1.5e-14
 */
/*							igamc()
 *
 *	Complemented incomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igamc();
 *
 * y = igamc( a, x );
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *
 *  igamc(a,x)   =   1 - igam(a,x)
 *
 *                            inf.
 *                              -
 *                     1       | |  -t  a-1
 *               =   -----     |   e   t   dt.
 *                    -      | |
 *                   | (a)    -
 *                             x
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 * ACCURACY:
 *
 * Tested at random a, x.
 *                a         x                      Relative error:
 * arithmetic   domain   domain     # trials      peak         rms
 *    IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
 *    IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15
 */

/*
Cephes Math Library Release 2.0:  April, 1987
Copyright 1985, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

#include <config.h>
#include "mconf.h"

extern double MACHEP, MAXLOG;
static double big = 4.503599627370496e15;
static double biginv = 2.22044604925031308085e-16;

double
igamc (double a, double x)
{
  double ans, ax, c, yc, r, t, y, z;
  double pk, pkm1, pkm2, qk, qkm1, qkm2;

  if ((x <= 0) || (a <= 0))
    return (1.0);

  if ((x < 1.0) || (x < a))
    return (1.0 - igam (a, x));

  ax = a * log (x) - x - lgam (a);
  if (ax < -MAXLOG) {
    mtherr ("igamc", MATHERR_UNDERFLOW);
    return (0.0);
  }
  ax = exp (ax);

/* continued fraction */
  y = 1.0 - a;
  z = x + y + 1.0;
  c = 0.0;
  pkm2 = 1.0;
  qkm2 = x;
  pkm1 = x + 1.0;
  qkm1 = z * x;
  ans = pkm1 / qkm1;

  do {
    c += 1.0;
    y += 1.0;
    z += 2.0;
    yc = y * c;
    pk = pkm1 * z - pkm2 * yc;
    qk = qkm1 * z - qkm2 * yc;
    if (qk != 0) {
      r = pk / qk;
      t = fabs ((ans - r) / r);
      ans = r;
    } else
      t = 1.0;
    pkm2 = pkm1;
    pkm1 = pk;
    qkm2 = qkm1;
    qkm1 = qk;
    if (fabs (pk) > big) {
      pkm2 *= biginv;
      pkm1 *= biginv;
      qkm2 *= biginv;
      qkm1 *= biginv;
    }
  }
  while (t > MACHEP);

  return (ans * ax);
}



/* left tail of incomplete gamma function:
 *
 *          inf.      k
 *   a  -x   -       x
 *  x  e     >   ----------
 *           -     -
 *          k=0   | (a+k+1)
 *
 */

double
igam (double a, double x)
{
  double ans, ax, c, r;

  if ((x <= 0) || (a <= 0))
    return (0.0);

  if ((x > 1.0) && (x > a))
    return (1.0 - igamc (a, x));

/* Compute  x**a * exp(-x) / gamma(a)  */
  ax = a * log (x) - x - lgam (a);
  if (ax < -MAXLOG) {
    mtherr ("igam", MATHERR_UNDERFLOW);
    return (0.0);
  }
  ax = exp (ax);

/* power series */
  r = a;
  c = 1.0;
  ans = 1.0;

  do {
    r += 1.0;
    c *= x / r;
    ans += c;
  }
  while (c / ans > MACHEP);

  return (ans * ax / a);
}