The R Project SVN R

/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998      Ross Ihaka
 *  Copyright (C) 1999-2000 The R Development Core Team
 *  Copyright (C) 2003-2004     The R Foundation
 *  based on AS 239 (C) 1988 Royal Statistical Society
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, write to the Free Software
 *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA.
 *
 *  SYNOPSIS
 *
 *  #include <Rmath.h>
 *  double pgamma(double x, double alph, double scale,
 *            int lower_tail, int log_p)
 *
 *  DESCRIPTION
 *
 *  This function computes the distribution function for the
 *  gamma distribution with shape parameter alph and scale parameter
 *  scale.  This is also known as the incomplete gamma function.
 *  See Abramowitz and Stegun (6.5.1) for example.
 *
 *  NOTES
 *
 *  This function is an adaptation of Algorithm 239 from the
 *  Applied Statistics Series.  The algorithm is faster than
 *  those by W. Fullerton in the FNLIB library and also the
 *  TOMS 542 alorithm of W. Gautschi.  It provides comparable
 *  accuracy to those algorithms and is considerably simpler.
 *
 *  REFERENCES
 *
 *  Algorithm AS 239, Incomplete Gamma Function
 *  Applied Statistics 37, 1988.
 */

#include "nmath.h"
#include "dpq.h"
/*----------- DEBUGGING -------------
 *  make CFLAGS='-DDEBUG_p -g -I/usr/local/include -I../include'
 */


double pgamma(double x, double alph, double scale, int lower_tail, int log_p)
{
    const double
    xbig = 1.0e+8,
    xlarge = 1.0e+37,

    /* normal approx. for alph > alphlimit */
    alphlimit = 1e5;/* was 1000. till R.1.8.x */

    double pn1, pn2, pn3, pn4, pn5, pn6, arg, a, b, c, an, osum, sum;
    long n;
    int pearson;

    /* check that we have valid values for x and alph */

#ifdef IEEE_754
    if (ISNAN(x) || ISNAN(alph) || ISNAN(scale))
    return x + alph + scale;
#endif
#ifdef DEBUG_p
    REprintf("pgamma(x=%4g, alph=%4g, scale=%4g): ",x,alph,scale);
#endif
    if(alph <= 0. || scale <= 0.)
    ML_ERR_return_NAN;

    x /= scale;
#ifdef DEBUG_p
    REprintf("-> x=%4g; ",x);
#endif
#ifdef IEEE_754
    if (ISNAN(x)) /* eg. original x = scale = Inf */
    return x;
#endif
    if (x <= 0.)
    return R_DT_0;

#define USE_PNORM \
    pn1 = sqrt(alph) * 3. * (pow(x/alph, 1./3.) + 1. / (9. * alph) - 1.); \
    return pnorm(pn1, 0., 1., lower_tail, log_p);

    if (alph > alphlimit) { /* use a normal approximation */
    USE_PNORM;
    }

    if (x > xbig * alph) {
    if (x > DBL_MAX * alph)
        /* if x is extremely large __compared to alph__ then return 1 */
        return R_DT_1;
    else { /* this only "helps" when log_p = TRUE */
        USE_PNORM;
    }
    }

    if (x <= 1. || x < alph) {

    pearson = 1;/* use pearson's series expansion. */

    arg = alph * log(x) - x - lgammafn(alph + 1.);
#ifdef DEBUG_p
    REprintf("Pearson  arg=%g ", arg);
#endif
    c = 1.;
    sum = 1.;
    a = alph;
    do {
        a += 1.;
        c *= x / a;
        sum += c;
    } while (c > DBL_EPSILON * sum);
    }
    else { /* x >= max( 1, alph) */

    pearson = 0;/* use a continued fraction expansion */

    arg = alph * log(x) - x - lgammafn(alph);
#ifdef DEBUG_p
    REprintf("Cont.Fract. arg=%g ", arg);
#endif
    a = 1. - alph;
    b = a + x + 1.;
    pn1 = 1.;
    pn2 = x;
    pn3 = x + 1.;
    pn4 = x * b;
    sum = pn3 / pn4;
    for (n = 1; ; n++) {
        a += 1.;/* =   n+1 -alph */
        b += 2.;/* = 2(n+1)-alph+x */
        an = a * n;
        pn5 = b * pn3 - an * pn1;
        pn6 = b * pn4 - an * pn2;
        if (fabs(pn6) > 0.) {
        osum = sum;
        sum = pn5 / pn6;
        if (fabs(osum - sum) <= DBL_EPSILON * fmin2(1., sum))
            break;
        }
        pn1 = pn3;
        pn2 = pn4;
        pn3 = pn5;
        pn4 = pn6;
        if (fabs(pn5) >= xlarge) {
        /* re-scale the terms in continued fraction if they are large */
#ifdef DEBUG_p
        REprintf(" [r] ");
#endif
        pn1 /= xlarge;
        pn2 /= xlarge;
        pn3 /= xlarge;
        pn4 /= xlarge;
        }
    }
    }

    arg += log(sum);

    lower_tail = (lower_tail == pearson);

    if (log_p && lower_tail)
    return(arg);
    /* else */
    /* sum = exp(arg); and return   if(lower_tail) sum  else 1-sum : */
    return (lower_tail) ? exp(arg) : (log_p ? R_Log1_Exp(arg) : -expm1(arg));
}