src/nmath/rgamma.c

/*
 *  Mathlib : A C Library of Special Functions
 *  Copyright (C) 1998 Ross Ihaka
 *  Copyright (C) 2000--2016 The R Core 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
 *  https://www.R-project.org/Licenses/
 *
 *  SYNOPSIS
 *
 *    #include <Rmath.h>
 *    double rgamma(double a, double scale);
 *
 *  DESCRIPTION
 *
 *    Random variates from the gamma distribution.
 *
 *  REFERENCES
 *
 *    [1] Shape parameter a >= 1.  Algorithm GD in:
 *
 *	  Ahrens, J.H. and Dieter, U. (1982).
 *	  Generating gamma variates by a modified
 *	  rejection technique.
 *	  Comm. ACM, 25, 47-54.
 *
 *
 *    [2] Shape parameter 0 < a < 1. Algorithm GS in:
 *
 *	  Ahrens, J.H. and Dieter, U. (1974).
 *	  Computer methods for sampling from gamma, beta,
 *	  poisson and binomial distributions.
 *	  Computing, 12, 223-246.
 *
 *  Input : a     = 'shape' = alpha,
 *          scale = 'scale' = 1/rate of the gamma distribution w/ mean E[.] = a * scale
 *  Output: a variate from the gamma(a, scale)-distribution
 */

#include "nmath.h"

#define repeat for(;;)

double rgamma(double a, double scale)
{
/* Constants : */
    const static double sqrt32 = 5.656854;
    const static double exp_m1 = 0.36787944117144232159;/* exp(-1) = 1/e */

    /* Coefficients q[k] - for q0 = sum(q[k]*a^(-k))
     * Coefficients a[k] - for q = q0+(t*t/2)*sum(a[k]*v^k)
     * Coefficients e[k] - for exp(q)-1 = sum(e[k]*q^k)
     */
    const static double q1 = 0.04166669;
    const static double q2 = 0.02083148;
    const static double q3 = 0.00801191;
    const static double q4 = 0.00144121;
    const static double q5 = -7.388e-5;
    const static double q6 = 2.4511e-4;
    const static double q7 = 2.424e-4;

    const static double a1 = 0.3333333;
    const static double a2 = -0.250003;
    const static double a3 = 0.2000062;
    const static double a4 = -0.1662921;
    const static double a5 = 0.1423657;
    const static double a6 = -0.1367177;
    const static double a7 = 0.1233795;

    /* State variables [FIXME for threading!] :*/
    static double aa = 0.;
    static double aaa = 0.;
    static double s, s2, d;    /* no. 1 (step 1) */
    static double q0, b, si, c;/* no. 2 (step 4) */

    double e, p, q, r, t, u, v, w, x, ret_val;

    if (ISNAN(a) || ISNAN(scale))
	ML_WARN_return_NAN;
    if (a <= 0.0 || scale <= 0.0) {
	if(scale == 0. || a == 0.) return 0.;
	ML_WARN_return_NAN;
    }
    if(!R_FINITE(a) || !R_FINITE(scale)) return ML_POSINF;

    if (a < 1.) { /* GS algorithm for parameters a < 1 */
	e = 1.0 + exp_m1 * a;
	repeat {
	    p = e * unif_rand();
	    if (p >= 1.0) {
		x = -log((e - p) / a);
		if (exp_rand() >= (1.0 - a) * log(x))
		    break;
	    } else {
		x = exp(log(p) / a);
		if (exp_rand() >= x)
		    break;
	    }
	}
	return scale * x;
    }

    /* --- a >= 1 : GD algorithm --- */

    /* Step 1: Recalculations of s2, s, d if a has changed */
    if (a != aa) {
	aa = a;
	s2 = a - 0.5;
	s = sqrt(s2);
	d = sqrt32 - s * 12.0;
    }
    /* Step 2: t = standard normal deviate,
               x = (s,1/2) -normal deviate. */

    /* immediate acceptance (i) */
    t = norm_rand();
    x = s + 0.5 * t;
    ret_val = x * x;
    if (t >= 0.0)
	return scale * ret_val;

    /* Step 3: u = 0,1 - uniform sample. squeeze acceptance (s) */
    u = unif_rand();
    if (d * u <= t * t * t)
	return scale * ret_val;

    /* Step 4: recalculations of q0, b, si, c if necessary */

    if (a != aaa) {
	aaa = a;
	r = 1.0 / a;
	q0 = ((((((q7 * r + q6) * r + q5) * r + q4) * r + q3) * r
	       + q2) * r + q1) * r;

	/* Approximation depending on size of parameter a */
	/* The constants in the expressions for b, si and c */
	/* were established by numerical experiments */

	if (a <= 3.686) {
	    b = 0.463 + s + 0.178 * s2;
	    si = 1.235;
	    c = 0.195 / s - 0.079 + 0.16 * s;
	} else if (a <= 13.022) {
	    b = 1.654 + 0.0076 * s2;
	    si = 1.68 / s + 0.275;
	    c = 0.062 / s + 0.024;
	} else {
	    b = 1.77;
	    si = 0.75;
	    c = 0.1515 / s;
	}
    }
    /* Step 5: no quotient test if x not positive */

    if (x > 0.0) {
	/* Step 6: calculation of v and quotient q */
	v = t / (s + s);
	if (fabs(v) <= 0.25)
	    q = q0 + 0.5 * t * t * ((((((a7 * v + a6) * v + a5) * v + a4) * v
				      + a3) * v + a2) * v + a1) * v;
	else
	    q = q0 - s * t + 0.25 * t * t + (s2 + s2) * log(1.0 + v);


	/* Step 7: quotient acceptance (q) */
	if (log(1.0 - u) <= q)
	    return scale * ret_val;
    }

    repeat {
	/* Step 8: e = standard exponential deviate
	 *	u =  0,1 -uniform deviate
	 *	t = (b,si)-double exponential (laplace) sample */
	e = exp_rand();
	u = unif_rand();
	u = u + u - 1.0;
	if (u < 0.0)
	    t = b - si * e;
	else
	    t = b + si * e;
	/* Step	 9:  rejection if t < tau(1) = -0.71874483771719 */
	if (t >= -0.71874483771719) {
	    /* Step 10:	 calculation of v and quotient q */
	    v = t / (s + s);
	    if (fabs(v) <= 0.25)
		q = q0 + 0.5 * t * t *
		    ((((((a7 * v + a6) * v + a5) * v + a4) * v + a3) * v
		      + a2) * v + a1) * v;
	    else
		q = q0 - s * t + 0.25 * t * t + (s2 + s2) * log(1.0 + v);
	    /* Step 11:	 hat acceptance (h) */
	    /* (if q not positive go to step 8) */
	    if (q > 0.0) {
		w = expm1(q);
		/*  ^^^^^ original code had approximation with rel.err < 2e-7 */
		/* if t is rejected sample again at step 8 */
		if (c * fabs(u) <= w * exp(e - 0.5 * t * t))
		    break;
	    }
	}
    } /* repeat .. until  `t' is accepted */
    x = s + 0.5 * t;
    return scale * x * x;
}