/*							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 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*/

#include "mconf.h"

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) {
#if 0
	mtherr("igamc", CEPHES_UNDERFLOW);
#endif
	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", CEPHES_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;
}
