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/*
* Mathlib : A C Library of Special Functions
* Copyright (C) 2005-6 Morten Welinder <terra@gnome.org>
* Copyright (C) 2005-6 The R Foundation
* Copyright (C) 2006 The R Core Development Team
*
* 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, a copy is available at
* http://www.r-project.org/Licenses/
*
* SYNOPSIS
*
* #include <Rmath.h>
*
* double pgamma (double x, double alph, double scale,
* int lower_tail, int log_p)
*
* double log1pmx (double x)
* double lgamma1p (double a)
*
* double logspace_add (double logx, double logy)
* double logspace_sub (double logx, double logy)
*
*
* 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
*
* Complete redesign by Morten Welinder, originally for Gnumeric.
* Improvements (e.g. "while NEEDED_SCALE") by Martin Maechler
* The old version can be activated by compiling with -DR_USE_OLD_PGAMMA
*
* REFERENCES
*
*/
#include "nmath.h"
#include "dpq.h"
/*----------- DEBUGGING -------------
* make CFLAGS='-DDEBUG_p -g -I/usr/local/include -I../include'
* (cd ~/R/D/r-devel/Linux-inst/src/nmath; gcc -std=gnu99 -I. -I../../src/include -I../../../R/src/include -I/usr/local/include -DDEBUG_p -g -O2 -c ../../../R/src/nmath/pgamma.c -o pgamma.o)
*/
/* Scalefactor:= (2^32)^8 = 2^256 = 1.157921e+77 */
#define SQR(x) ((x)*(x))
static const double scalefactor = SQR(SQR(SQR(4294967296.0)));
#undef SQR
/* If |x| > |k| * M_cutoff, then log[ exp(-x) * k^x ] =~= -x */
static const double M_cutoff = M_LN2 * DBL_MAX_EXP / DBL_EPSILON;/*=3.196577e18*/
/* Continued fraction for calculation of
* 1/i + x/(i+d) + x^2/(i+2*d) + x^3/(i+3*d) + ... = sum_{k=0}^Inf x^k/(i+k*d)
*
* auxilary in log1pmx() and lgamma1p()
*/
static double
logcf (double x, double i, double d,
double eps /* ~ relative tolerance */)
{
double c1 = 2 * d;
double c2 = i + d;
double c4 = c2 + d;
double a1 = c2;
double b1 = i * (c2 - i * x);
double b2 = d * d * x;
double a2 = c4 * c2 - b2;
#if 0
assert (i > 0);
assert (d >= 0);
#endif
b2 = c4 * b1 - i * b2;
while (fabs(a2 * b1 - a1 * b2) > fabs(eps * b1 * b2)) {
double c3 = c2*c2*x;
c2 += d;
c4 += d;
a1 = c4 * a2 - c3 * a1;
b1 = c4 * b2 - c3 * b1;
c3 = c1 * c1 * x;
c1 += d;
c4 += d;
a2 = c4 * a1 - c3 * a2;
b2 = c4 * b1 - c3 * b2;
if (fabs (b2) > scalefactor) {
a1 /= scalefactor;
b1 /= scalefactor;
a2 /= scalefactor;
b2 /= scalefactor;
} else if (fabs (b2) < 1 / scalefactor) {
a1 *= scalefactor;
b1 *= scalefactor;
a2 *= scalefactor;
b2 *= scalefactor;
}
}
return a2 / b2;
}
/* Accurate calculation of log(1+x)-x, particularly for small x. */
double log1pmx (double x)
{
static const double minLog1Value = -0.79149064;
if (x > 1 || x < minLog1Value)
return log1p(x) - x;
else { /* -.791 <= x <= 1 -- expand in [x/(2+x)]^2 =: y :
* log(1+x) - x = x/(2+x) * [ 2 * y * S(y) - x], with
* ---------------------------------------------
* S(y) = 1/3 + y/5 + y^2/7 + ... = \sum_{k=0}^\infty y^k / (2k + 3)
*/
double r = x / (2 + x), y = r * r;
if (fabs(x) < 1e-2) {
static const double two = 2;
return r * ((((two / 9 * y + two / 7) * y + two / 5) * y +
two / 3) * y - x);
} else {
static const double tol_logcf = 1e-14;
return r * (2 * y * logcf (y, 3, 2, tol_logcf) - x);
}
}
}
/* Compute log(gamma(a+1)) accurately also for small a (0 < a < 0.5). */
double lgamma1p (double a)
{
const double eulers_const = 0.5772156649015328606065120900824024;
/* coeffs[i] holds (zeta(i+2)-1)/(i+2) , i = 0:(N-1), N = 40 : */
const int N = 40;
static const double coeffs[40] = {
0.3224670334241132182362075833230126e-0,/* = (zeta(2)-1)/2 */
0.6735230105319809513324605383715000e-1,/* = (zeta(3)-1)/3 */
0.2058080842778454787900092413529198e-1,
0.7385551028673985266273097291406834e-2,
0.2890510330741523285752988298486755e-2,
0.1192753911703260977113935692828109e-2,
0.5096695247430424223356548135815582e-3,
0.2231547584535793797614188036013401e-3,
0.9945751278180853371459589003190170e-4,
0.4492623673813314170020750240635786e-4,
0.2050721277567069155316650397830591e-4,
0.9439488275268395903987425104415055e-5,
0.4374866789907487804181793223952411e-5,
0.2039215753801366236781900709670839e-5,
0.9551412130407419832857179772951265e-6,
0.4492469198764566043294290331193655e-6,
0.2120718480555466586923135901077628e-6,
0.1004322482396809960872083050053344e-6,
0.4769810169363980565760193417246730e-7,
0.2271109460894316491031998116062124e-7,
0.1083865921489695409107491757968159e-7,
0.5183475041970046655121248647057669e-8,
0.2483674543802478317185008663991718e-8,
0.1192140140586091207442548202774640e-8,
0.5731367241678862013330194857961011e-9,
0.2759522885124233145178149692816341e-9,
0.1330476437424448948149715720858008e-9,
0.6422964563838100022082448087644648e-10,
0.3104424774732227276239215783404066e-10,
0.1502138408075414217093301048780668e-10,
0.7275974480239079662504549924814047e-11,
0.3527742476575915083615072228655483e-11,
0.1711991790559617908601084114443031e-11,
0.8315385841420284819798357793954418e-12,
0.4042200525289440065536008957032895e-12,
0.1966475631096616490411045679010286e-12,
0.9573630387838555763782200936508615e-13,
0.4664076026428374224576492565974577e-13,
0.2273736960065972320633279596737272e-13,
0.1109139947083452201658320007192334e-13/* = (zeta(40+1)-1)/(40+1) */
};
const double c = 0.2273736845824652515226821577978691e-12;/* zeta(N+2)-1 */
const double tol_logcf = 1e-14;
double lgam;
int i;
if (fabs (a) >= 0.5)
return lgammafn (a + 1);
/* Abramowitz & Stegun 6.1.33 : for |x| < 2,
* <==> log(gamma(1+x)) = -(log(1+x) - x) - gamma*x + x^2 * \sum_{n=0}^\infty c_n (-x)^n
* where c_n := (Zeta(n+2) - 1)/(n+2) = coeffs[n]
*
* Here, another convergence acceleration trick is used to compute
* lgam(x) := sum_{n=0..Inf} c_n (-x)^n
*/
lgam = c * logcf(-a / 2, N + 2, 1, tol_logcf);
for (i = N - 1; i >= 0; i--)
lgam = coeffs[i] - a * lgam;
return (a * lgam - eulers_const) * a - log1pmx (a);
} /* lgamma1p */
/*
* Compute the log of a sum from logs of terms, i.e.,
*
* log (exp (logx) + exp (logy))
*
* without causing overflows and without throwing away large handfuls
* of accuracy.
*/
double logspace_add (double logx, double logy)
{
return fmax2 (logx, logy) + log1p (exp (-fabs (logx - logy)));
}
/*
* Compute the log of a difference from logs of terms, i.e.,
*
* log (exp (logx) - exp (logy))
*
* without causing overflows and without throwing away large handfuls
* of accuracy.
*/
double logspace_sub (double logx, double logy)
{
return logx + log1p (-exp (logy - logx));
}
#ifndef R_USE_OLD_PGAMMA
/* dpois_wrap (x_P_1, lambda, g_log) ==
* dpois (x_P_1 - 1, lambda, g_log) := exp(-L) L^k / gamma(k+1) , k := x_P_1 - 1
*/
static double
dpois_wrap (double x_plus_1, double lambda, int give_log)
{
#ifdef DEBUG_p
JREprintf (" dpois_wrap(x+1=%.14g, lambda=%.14g, log=%d)\n",
x_plus_1, lambda, give_log);
#endif
if (!R_FINITE(lambda))
return R_D__0;
if (x_plus_1 > 1)
return dpois_raw (x_plus_1 - 1, lambda, give_log);
if (lambda > fabs(x_plus_1 - 1) * M_cutoff)
return R_D_exp(-lambda - lgammafn(x_plus_1));
else {
double d = dpois_raw (x_plus_1, lambda, give_log);
#ifdef DEBUG_p
JREprintf (" -> d=dpois_raw(..)=%.14g\n", d);
#endif
return give_log
? d + log (x_plus_1 / lambda)
: d * (x_plus_1 / lambda);
}
}
/*
* Abramowitz and Stegun 6.5.29 [right]
*/
static double
pgamma_smallx (double x, double alph, int lower_tail, int log_p)
{
double sum = 0, c = alph, n = 0, term;
#ifdef DEBUG_p
JREprintf (" pg_smallx(x=%.12g, alph=%.12g): ", x, alph);
#endif
/*
* Relative to 6.5.29 all terms have been multiplied by alph
* and the first, thus being 1, is omitted.
*/
do {
n++;
c *= -x / n;
term = c / (alph + n);
sum += term;
} while (fabs (term) > DBL_EPSILON * fabs (sum));
#ifdef DEBUG_p
JREprintf (" %d terms --> conv.sum=%g;", n, sum);
#endif
if (lower_tail) {
double f1 = log_p ? log1p (sum) : 1 + sum;
double f2;
if (alph > 1) {
f2 = dpois_raw (alph, x, log_p);
f2 = log_p ? f2 + x : f2 * exp (x);
} else if (log_p)
f2 = alph * log (x) - lgamma1p (alph);
else
f2 = pow (x, alph) / exp (lgamma1p (alph));
#ifdef DEBUG_p
JREprintf (" (f1,f2)= (%g,%g)\n", f1,f2);
#endif
return log_p ? f1 + f2 : f1 * f2;
} else {
double lf2 = alph * log (x) - lgamma1p (alph);
#ifdef DEBUG_p
JREprintf (" 1:%.14g 2:%.14g\n", alph * log (x), lgamma1p (alph));
JREprintf (" sum=%.14g log(1+sum)=%.14g lf2=%.14g\n",
sum, log1p (sum), lf2);
#endif
if (log_p)
return R_Log1_Exp (log1p (sum) + lf2);
else {
double f1m1 = sum;
double f2m1 = expm1 (lf2);
return -(f1m1 + f2m1 + f1m1 * f2m1);
}
}
} /* pgamma_smallx() */
static double
pd_upper_series (double x, double y, int log_p)
{
double term = x / y;
double sum = term;
do {
y++;
term *= x / y;
sum += term;
} while (term > sum * DBL_EPSILON);
/* sum = \sum_{n=1}^ oo x^n / (y*(y+1)*...*(y+n-1))
* = \sum_{n=0}^ oo x^(n+1) / (y*(y+1)*...*(y+n))
* = x/y * (1 + \sum_{n=1}^oo x^n / ((y+1)*...*(y+n)))
* ~ x/y + o(x/y) {which happens when alph -> Inf}
*/
return log_p ? log (sum) : sum;
}
/* Continued fraction for calculation of
* ???
* = (y / d) + o(y/d)
*/
static double
pd_lower_cf (double y, double d)
{
double f = 0, of;
double i, c2, c3, c4, a1, b1, a2, b2;
#define NEEDED_SCALE \
(b2 > scalefactor) { \
a1 /= scalefactor; \
b1 /= scalefactor; \
a2 /= scalefactor; \
b2 /= scalefactor; \
}
#define max_it 200000
#ifdef DEBUG_p
JREprintf("pd_lower_cf(y=%.14g, d=%.14g)", y, d);
#endif
if (y == 0 || (R_FINITE(y) && !R_FINITE(d))) /* includes d = Inf or y = 0 */
return 0;
/* Needed, e.g. for pgamma(10^c(100,295), shape= 1.1, log=TRUE) */
if(fabs(y - 1) < fabs(d) * 1e-20) {
#ifdef DEBUG_p
JREprintf(" very small 'y' -> returning (y/d)\n");
#endif
return (y/d);
}
c2 = y;
c4 = d; /* original (y,d), *not* potentially scaled ones!*/
a1 = 0; b1 = 1;
a2 = y; b2 = d;
while NEEDED_SCALE
/* if(a2 == 0) return 0;/\* just in case, e.g. d=y=0 *\/ */
i = 0;
while (i < max_it) {
i++; c2--; c3 = i * c2; c4 += 2;
/* c2 = y - i, c3 = i(y - i), c4 = d + 2i, for i odd */
a1 = c4 * a2 + c3 * a1;
b1 = c4 * b2 + c3 * b1;
i++; c2--; c3 = i * c2; c4 += 2;
/* c2 = y - i, c3 = i(y - i), c4 = d + 2i, for i even */
a2 = c4 * a1 + c3 * a2;
b2 = c4 * b1 + c3 * b2;
if NEEDED_SCALE
if (b2 != 0) {
of = f;
f = a2 / b2;
#ifdef UP_TO_2009_11_07__NO_LONGER
/* convergence check: relative; absolute for small f : */
if (fabs (f - of) <= DBL_EPSILON * fmax2(1., fabs(f))) { .. }
#endif
/* convergence check */
if (fabs(f - of) <= DBL_EPSILON * fabs(f)) {
#ifdef DEBUG_p
JREprintf(" %g iter.\n", i);
#endif
return f;
}
}
}
MATHLIB_WARNING(" ** NON-convergence in pgamma()'s pd_lower_cf() f= %g.\n",
f);
return f;/* should not happen ... */
} /* pd_lower_cf() */
#undef NEEDED_SCALE
static double
pd_lower_series (double lambda, double y)
{
double term = 1, sum = 0;
#ifdef DEBUG_p
JREprintf("pd_lower_series(lam=%.14g, y=%.14g) ...", lambda, y);
#endif
while (y >= 1 && term > sum * DBL_EPSILON) {
term *= y / lambda;
sum += term;
y--;
}
/* sum = \sum_{n=0}^ oo y*(y-1)*...*(y - n) / lambda^(n+1)
* = y/lambda * (1 + \sum_{n=1}^Inf (y-1)*...*(y-n) / lambda^n)
* ~ y/lambda + o(y/lambda)
*/
#ifdef DEBUG_p
JREprintf(" done: term=%g, sum=%g, y= %g\n", term, sum, y);
#endif
if (y != floor (y)) {
/*
* The series does not converge as the terms start getting
* bigger (besides flipping sign) for y < -lambda.
*/
double f;
#ifdef DEBUG_p
JREprintf(" y not int: add another term ");
#endif
/* FIXME: in quite few cases, adding term*f has no effect (f too small)
* and is unnecessary e.g. for pgamma(4e12, 121.1) */
f = pd_lower_cf (y, lambda + 1 - y);
#ifdef DEBUG_p
JREprintf(" (= %.14g) * term = %.14g to sum %g\n", f, term * f, sum);
#endif
sum += term * f;
}
return sum;
} /* pd_lower_series() */
/*
* Compute the following ratio with higher accuracy that would be had
* from doing it directly.
*
* dnorm (x, 0, 1, FALSE)
* ----------------------------------
* pnorm (x, 0, 1, lower_tail, FALSE)
*
* Abramowitz & Stegun 26.2.12
*/
static double
dpnorm (double x, int lower_tail, double lp)
{
/*
* So as not to repeat a pnorm call, we expect
*
* lp == pnorm (x, 0, 1, lower_tail, TRUE)
*
* but use it only in the non-critical case where either x is small
* or p==exp(lp) is close to 1.
*/
if (x < 0) {
x = -x;
lower_tail = !lower_tail;
}
if (x > 10 && !lower_tail) {
double term = 1 / x;
double sum = term;
double x2 = x * x;
double i = 1;
do {
term *= -i / x2;
sum += term;
i += 2;
} while (fabs (term) > DBL_EPSILON * sum);
return 1 / sum;
} else {
double d = dnorm (x, 0, 1, FALSE);
return d / exp (lp);
}
}
/*
* Asymptotic expansion to calculate the probability that Poisson variate
* has value <= x.
* Various assertions about this are made (without proof) at
* http://members.aol.com/iandjmsmith/PoissonApprox.htm
*/
static double
ppois_asymp (double x, double lambda, int lower_tail, int log_p)
{
static const double coefs_a[8] = {
-1e99, /* placeholder used for 1-indexing */
2/3.,
-4/135.,
8/2835.,
16/8505.,
-8992/12629925.,
-334144/492567075.,
698752/1477701225.
};
static const double coefs_b[8] = {
-1e99, /* placeholder */
1/12.,
1/288.,
-139/51840.,
-571/2488320.,
163879/209018880.,
5246819/75246796800.,
-534703531/902961561600.
};
double elfb, elfb_term;
double res12, res1_term, res1_ig, res2_term, res2_ig;
double dfm, pt_, s2pt, f, np;
int i;
dfm = lambda - x;
/* If lambda is large, the distribution is highly concentrated
about lambda. So representation error in x or lambda can lead
to arbitrarily large values of pt_ and hence divergence of the
coefficients of this approximation.
*/
pt_ = - log1pmx (dfm / x);
s2pt = sqrt (2 * x * pt_);
if (dfm < 0) s2pt = -s2pt;
res12 = 0;
res1_ig = res1_term = sqrt (x);
res2_ig = res2_term = s2pt;
for (i = 1; i < 8; i++) {
res12 += res1_ig * coefs_a[i];
res12 += res2_ig * coefs_b[i];
res1_term *= pt_ / i ;
res2_term *= 2 * pt_ / (2 * i + 1);
res1_ig = res1_ig / x + res1_term;
res2_ig = res2_ig / x + res2_term;
}
elfb = x;
elfb_term = 1;
for (i = 1; i < 8; i++) {
elfb += elfb_term * coefs_b[i];
elfb_term /= x;
}
if (!lower_tail) elfb = -elfb;
#ifdef DEBUG_p
JREprintf ("res12 = %.14g elfb=%.14g\n", elfb, res12);
#endif
f = res12 / elfb;
np = pnorm (s2pt, 0.0, 1.0, !lower_tail, log_p);
if (log_p) {
double n_d_over_p = dpnorm (s2pt, !lower_tail, np);
#ifdef DEBUG_p
JREprintf ("pp*_asymp(): f=%.14g np=e^%.14g nd/np=%.14g f*nd/np=%.14g\n",
f, np, n_d_over_p, f * n_d_over_p);
#endif
return np + log1p (f * n_d_over_p);
} else {
double nd = dnorm (s2pt, 0., 1., log_p);
#ifdef DEBUG_p
JREprintf ("pp*_asymp(): f=%.14g np=%.14g nd=%.14g f*nd=%.14g\n",
f, np, nd, f * nd);
#endif
return np + f * nd;
}
} /* ppois_asymp() */
double pgamma_raw (double x, double alph, int lower_tail, int log_p)
{
/* Here, assume that (x,alph) are not NA & alph > 0 . */
double res;
#ifdef DEBUG_p
JREprintf("pgamma_raw(x=%.14g, alph=%.14g, low=%d, log=%d)\n",
x, alph, lower_tail, log_p);
#endif
R_P_bounds_01(x, 0., ML_POSINF);
if (x < 1) {
res = pgamma_smallx (x, alph, lower_tail, log_p);
} else if (x <= alph - 1 && x < 0.8 * (alph + 50)) {
/* incl. large alph compared to x */
double sum = pd_upper_series (x, alph, log_p);/* = x/alph + o(x/alph) */
double d = dpois_wrap (alph, x, log_p);
#ifdef DEBUG_p
JREprintf(" alph 'large': sum=pd_upper*()= %.12g, d=dpois_w(*)= %.12g\n",
sum, d);
#endif
if (!lower_tail)
res = log_p
? R_Log1_Exp (d + sum)
: 1 - d * sum;
else
res = log_p ? sum + d : sum * d;
} else if (alph - 1 < x && alph < 0.8 * (x + 50)) {
/* incl. large x compared to alph */
double sum;
double d = dpois_wrap (alph, x, log_p);
#ifdef DEBUG_p
JREprintf(" x 'large': d=dpois_w(*)= %.14g ", d);
#endif
if (alph < 1) {
if (x * DBL_EPSILON > 1 - alph)
sum = R_D__1;
else {
double f = pd_lower_cf (alph, x - (alph - 1)) * x / alph;
/* = [alph/(x - alph+1) + o(alph/(x-alph+1))] * x/alph = 1 + o(1) */
sum = log_p ? log (f) : f;
}
} else {
sum = pd_lower_series (x, alph - 1);/* = (alph-1)/x + o((alph-1)/x) */
sum = log_p ? log1p (sum) : 1 + sum;
}
#ifdef DEBUG_p
JREprintf(", sum= %.14g\n", sum);
#endif
if (!lower_tail)
res = log_p ? sum + d : sum * d;
else
res = log_p
? R_Log1_Exp (d + sum)
: 1 - d * sum;
} else { /* x >= 1 and x fairly near alph. */
#ifdef DEBUG_p
JREprintf(" using ppois_asymp()\n");
#endif
res = ppois_asymp (alph - 1, x, !lower_tail, log_p);
}
/*
* We lose a fair amount of accuracy to underflow in the cases
* where the final result is very close to DBL_MIN. In those
* cases, simply redo via log space.
*/
if (!log_p && res < DBL_MIN / DBL_EPSILON) {
/* with(.Machine, double.xmin / double.eps) #|-> 1.002084e-292 */
#ifdef DEBUG_p
JREprintf(" very small res=%.14g; -> recompute via log\n", res);
#endif
return exp (pgamma_raw (x, alph, lower_tail, 1));
} else
return res;
}
double pgamma(double x, double alph, double scale, int lower_tail, int log_p)
{
#ifdef IEEE_754
if (ISNAN(x) || ISNAN(alph) || ISNAN(scale))
return x + alph + scale;
#endif
if(alph < 0. || scale <= 0.)
ML_ERR_return_NAN;
x /= scale;
#ifdef IEEE_754
if (ISNAN(x)) /* eg. original x = scale = +Inf */
return x;
#endif
if(alph == 0.) /* limit case; useful e.g. in pnchisq() */
return (x < 0) ? R_DT_0: R_DT_1;
return pgamma_raw (x, alph, lower_tail, log_p);
}
/* From: terra@gnome.org (Morten Welinder)
* To: R-bugs@biostat.ku.dk
* Cc: maechler@stat.math.ethz.ch
* Subject: Re: [Rd] pgamma discontinuity (PR#7307)
* Date: Tue, 11 Jan 2005 13:57:26 -0500 (EST)
* this version of pgamma appears to be quite good and certainly a vast
* improvement over current R code. (I last looked at 2.0.1) Apart from
* type naming, this is what I have been using for Gnumeric 1.4.1.
* This could be included into R as-is, but you might want to benefit from
* making logcf, log1pmx, lgamma1p, and possibly logspace_add/logspace_sub
* available to other parts of R.
* MM: I've not (yet?) taken logcf(), but the other four
*/
#else
/* R_USE_OLD_PGAMMA */
/*
* 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
*
* ................
*
* 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.
*/
/* now would need this here: */
double attribute_hidden pgamma_raw(x, alph, lower_tail, log_p) {
return pgamma(x, alph, 1, lower_tail, log_p);
}
double pgamma(double x, double alph, double scale, int lower_tail, int log_p)
{
const static 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
JREprintf("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
JREprintf("-> 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
JREprintf("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
JREprintf("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
JREprintf(" [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));
}
#endif
/* R_USE_OLD_PGAMMA */