/*
* Mathlib : A C Library of Special Functions
* Copyright (C) 1998 Ross Ihaka
* Copyright (C) 2000 The R Development Core Team
* Copyright (C) 2004 The R Foundation
* based on AS 91 (C) 1979 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301
* USA.
*
* DESCRIPTION
*
* Compute the quantile function of the gamma distribution.
*
* NOTES
*
* This function is based on the Applied Statistics
* Algorithm AS 91 ("ppchi2") and via pgamma(.) AS 239.
*
* R core improvements:
* o lower_tail, log_p
* o non-trivial result for p outside [0.000002, 0.999998]
* o p ~ 1 no longer gives +Inf; final Newton step(s)
*
* REFERENCES
*
* Best, D. J. and D. E. Roberts (1975).
* Percentage Points of the Chi-Squared Distribution.
* Applied Statistics 24, page 385. */
#include "nmath.h"
#include "dpq.h"
#ifdef DEBUG_qgamma
# define DEBUG_q
#endif
attribute_hidden
double qchisq_appr(double p, double nu, double g/* = log Gamma(nu/2) */,
int lower_tail, int log_p, double tol /* EPS1 */)
{
#define C7 4.67
#define C8 6.66
#define C9 6.73
#define C10 13.32
double p_, alpha, a, c, ch, p1;
double p2, q, t, x;
/* test arguments and initialise */
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(nu))
return p + nu;
#endif
R_Q_P01_check(p);
if (nu <= 0) ML_ERR_return_NAN;
p_ = R_DT_qIv(p);/* lower_tail prob (in any case) */
alpha = 0.5 * nu;/* = [pq]gamma() shape */
c = alpha-1;
if(nu < (-1.24)*(p1 = R_DT_log(p))) { /* for small chi-squared */
ch = exp((log(alpha) + p1 + g)/alpha + M_LN2);
#ifdef DEBUG_qgamma
REprintf(" small chi-sq., ch0 = %g\n", ch);
#endif
} else if(nu > 0.32) { /* using Wilson and Hilferty estimate */
x = qnorm(p, 0, 1, lower_tail, log_p);
p1 = 2./(9*nu);
ch = nu*pow(x*sqrt(p1) + 1-p1, 3);
#ifdef DEBUG_qgamma
REprintf(" nu > .32: Wilson-Hilferty; x = %7g\n", x);
#endif
/* approximation for p tending to 1: */
if( ch > 2.2*nu + 6 )
ch = -2*(R_DT_Clog(p) - c*log(0.5*ch) + g);
} else { /* "small nu" : 1.24*(-log(p)) <= nu <= 0.32 */
ch = 0.4;
a = R_DT_Clog(p) + g + c*M_LN2;
#ifdef DEBUG_qgamma
REprintf(" nu <= .32: a = %7g\n", a);
#endif
do {
q = ch;
p1 = 1. / (1+ch*(C7+ch));
p2 = ch*(C9+ch*(C8+ch));
t = -0.5 +(C7+2*ch)*p1 - (C9+ch*(C10+3*ch))/p2;
ch -= (1- exp(a+0.5*ch)*p2*p1)/t;
} while(fabs(q - ch) > tol * fabs(ch));
}
return ch;
}
double qgamma(double p, double alpha, double scale, int lower_tail, int log_p)
/* shape = alpha */
{
#define EPS1 1e-2
#define EPS2 5e-7/* final precision of AS 91 */
#define EPS_N 1e-15/* precision of Newton step / iterations */
#define MAXIT 1000/* was 20 */
#define pMIN 1e-100 /* was 0.000002 = 2e-6 */
#define pMAX (1-1e-14)/* was (1-1e-12) and 0.999998 = 1 - 2e-6 */
const static double
i420 = 1./ 420.,
i2520 = 1./ 2520.,
i5040 = 1./ 5040;
double p_, a, b, c, g, ch, ch0, p1;
double p2, q, s1, s2, s3, s4, s5, s6, t, x;
int i, max_it_Newton = 1;
/* test arguments and initialise */
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(alpha) || ISNAN(scale))
return p + alpha + scale;
#endif
R_Q_P01_boundaries(p, 0., ML_POSINF);
if (alpha <= 0 || scale <= 0) ML_ERR_return_NAN;
p_ = R_DT_qIv(p);/* lower_tail prob (in any case) */
#ifdef DEBUG_qgamma
REprintf("qgamma(p=%7g, alpha=%7g, scale=%7g, l.t.=%2d, log_p=%2d): ",
p,alpha,scale, lower_tail, log_p);
#endif
g = lgammafn(alpha);/* log Gamma(v/2) */
/*----- Phase I : Starting Approximation */
ch = qchisq_appr(p, /* nu= 'df' = */ 2*alpha, /* lgamma(nu/2)= */ g,
lower_tail, log_p, /* tol= */ EPS1);
if(!R_FINITE(ch)) {
/* forget about all iterations! */
max_it_Newton = 0; goto END;
}
if(ch < EPS2) {/* Corrected according to AS 91; MM, May 25, 1999 */
max_it_Newton = 20;
goto END;/* and do Newton steps */
}
/* FIXME: This (cutoff to {0, +Inf}) is far from optimal
* ----- when log_p or !lower_tail : */
if(p_ > pMAX || p_ < pMIN) {
/* did return ML_POSINF or 0.; much better: */
max_it_Newton = 20;
goto END;/* and do Newton steps */
}
#ifdef DEBUG_qgamma
REprintf("\t==> ch = %10g:", ch);
#endif
/*----- Phase II: Iteration
* Call pgamma() [AS 239] and calculate seven term taylor series
*/
c = alpha-1;
s6 = (120+c*(346+127*c)) * i5040; /* used below, is "const" */
ch0 = ch;/* save initial approx. */
for(i=1; i <= MAXIT; i++ ) {
q = ch;
p1 = 0.5*ch;
p2 = p_ - pgamma_raw(p1, alpha, /*lower_tail*/TRUE, /*log_p*/FALSE);
#ifdef DEBUG_qgamma
if(i == 1) REprintf(" Ph.II iter; ch=%g, p2=%g\n", ch, p2);
if(i >= 2) REprintf(" it=%d, ch=%g, p2=%g\n", i, p2);
#endif
#ifdef IEEE_754
if(!R_FINITE(p2))
#else
if(errno != 0)
#endif
{ ch = ch0; max_it_Newton = 27; goto END; }/*was return ML_NAN;*/
t = p2*exp(alpha*M_LN2+g+p1-c*log(ch));
b = t/ch;
a = 0.5*t - b*c;
s1 = (210+ a*(140+a*(105+a*(84+a*(70+60*a))))) * i420;
s2 = (420+ a*(735+a*(966+a*(1141+1278*a)))) * i2520;
s3 = (210+ a*(462+a*(707+932*a))) * i2520;
s4 = (252+ a*(672+1182*a) + c*(294+a*(889+1740*a))) * i5040;
s5 = (84+2264*a + c*(1175+606*a)) * i2520;
ch += t*(1+0.5*t*s1-b*c*(s1-b*(s2-b*(s3-b*(s4-b*(s5-b*s6))))));
if(fabs(q - ch) < EPS2*ch)
goto END;
}
/* no convergence in MAXIT iterations -- but we add Newton now... */
#ifdef DEBUG_q
MATHLIB_WARNING3("qgamma(%g) not converged in %d iterations; rel.ch=%g\n",
p, MAXIT, ch/fabs(q - ch));
#endif
/* was
* ML_ERROR(ME_PRECISION, "qgamma");
* does nothing in R !*/
END:
/* PR# 2214 : From: Morten Welinder <terra@diku.dk>, Fri, 25 Oct 2002 16:50
-------- To: R-bugs@biostat.ku.dk Subject: qgamma precision
* With a final Newton step, double accuracy, e.g. for (p= 7e-4; nu= 0.9)
*
* Improved (MM): - only if rel.Err > EPS_N (= 1e-15);
* - also for lower_tail = FALSE or log_p = TRUE
* - optionally *iterate* Newton
*/
x = 0.5*scale*ch;
if(max_it_Newton)
p_ = pgamma(x, alpha, scale, lower_tail, log_p);
for(i = 1; i <= max_it_Newton; i++) {
p1 = p_ - p;
#ifdef DEBUG_qgamma
if(i == 1) REprintf("\n it=%d: p=%g, x = %g, p.=%g; p1:=D{p}=%g\n",
i, p, x, p_, p1);
if(i >= 2) REprintf(" it=%d, d{p}=%g\n", i, p1);
#endif
if(fabs(p1) < fabs(EPS_N * p))
break;
/* else */
if((g = dgamma(x, alpha, scale, log_p)) == R_D__0) {
#ifdef DEBUG_q
if(i == 1) REprintf("no final Newton step because dgamma(*)== 0!\n");
#endif
break;
}
/* else :
* delta x = f(x)/f'(x);
* if(log_p) f(x) := log P(x) - p; f'(x) = d/dx log P(x) = P' / P
* ==> f(x)/f'(x) = f*P / P' = f*exp(p_) / P' (since p_ = log P(x))
*/
t = log_p ? p1*exp(p_ - g) : p1/g ;/* = "delta x" */
t = lower_tail ? x - t : x + t;
p_ = pgamma (t, alpha, scale, lower_tail, log_p);
if (fabs(p_ - p) > fabs(p1) ||
(i > 1 && fabs(p_ - p) == fabs(p1)) /* <- against flip-flop */) {
/* no improvement */
#ifdef DEBUG_q
if(i == 1 && max_it_Newton > 1)
REprintf("no Newton step done since delta{p} >= last delta\n");
#endif
break;
} /* else : */
x = t;
}
return x;
}
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