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/* randist/gamma.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 James Theiler, Brian Gough
*
* 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.
*/
#include <config.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
static double gamma_large (const gsl_rng * r, const double a);
static double gamma_frac (const gsl_rng * r, const double a);
/* The Gamma distribution of order a>0 is defined by:
p(x) dx = {1 / \Gamma(a) b^a } x^{a-1} e^{-x/b} dx
for x>0. If X and Y are independent gamma-distributed random
variables of order a1 and a2 with the same scale parameter b, then
X+Y has gamma distribution of order a1+a2.
The algorithms below are from Knuth, vol 2, 2nd ed, p. 129. */
double
gsl_ran_gamma_knuth (const gsl_rng * r, const double a, const double b)
{
/* assume a > 0 */
unsigned int na = floor (a);
if (a == na)
{
return b * gsl_ran_gamma_int (r, na);
}
else if (na == 0)
{
return b * gamma_frac (r, a);
}
else
{
return b * (gsl_ran_gamma_int (r, na) + gamma_frac (r, a - na)) ;
}
}
double
gsl_ran_gamma_int (const gsl_rng * r, const unsigned int a)
{
if (a < 12)
{
unsigned int i;
double prod = 1;
for (i = 0; i < a; i++)
{
prod *= gsl_rng_uniform_pos (r);
}
/* Note: for 12 iterations we are safe against underflow, since
the smallest positive random number is O(2^-32). This means
the smallest possible product is 2^(-12*32) = 10^-116 which
is within the range of double precision. */
return -log (prod);
}
else
{
return gamma_large (r, (double) a);
}
}
static double
gamma_large (const gsl_rng * r, const double a)
{
/* Works only if a > 1, and is most efficient if a is large
This algorithm, reported in Knuth, is attributed to Ahrens. A
faster one, we are told, can be found in: J. H. Ahrens and
U. Dieter, Computing 12 (1974) 223-246. */
double sqa, x, y, v;
sqa = sqrt (2 * a - 1);
do
{
do
{
y = tan (M_PI * gsl_rng_uniform (r));
x = sqa * y + a - 1;
}
while (x <= 0);
v = gsl_rng_uniform (r);
}
while (v > (1 + y * y) * exp ((a - 1) * log (x / (a - 1)) - sqa * y));
return x;
}
static double
gamma_frac (const gsl_rng * r, const double a)
{
/* This is exercise 16 from Knuth; see page 135, and the solution is
on page 551. */
double p, q, x, u, v;
p = M_E / (a + M_E);
do
{
u = gsl_rng_uniform (r);
v = gsl_rng_uniform_pos (r);
if (u < p)
{
x = exp ((1 / a) * log (v));
q = exp (-x);
}
else
{
x = 1 - log (v);
q = exp ((a - 1) * log (x));
}
}
while (gsl_rng_uniform (r) >= q);
return x;
}
double
gsl_ran_gamma_pdf (const double x, const double a, const double b)
{
if (x < 0)
{
return 0 ;
}
else if (x == 0)
{
if (a == 1)
return 1/b ;
else
return 0 ;
}
else if (a == 1)
{
return exp(-x/b)/b ;
}
else
{
double p;
double lngamma = gsl_sf_lngamma (a);
p = exp ((a - 1) * log (x/b) - x/b - lngamma)/b;
return p;
}
}
/* New version based on Marsaglia and Tsang, "A Simple Method for
* generating gamma variables", ACM Transactions on Mathematical
* Software, Vol 26, No 3 (2000), p363-372.
*
* Implemented by J.D.Lamb@btinternet.com, minor modifications for GSL
* by Brian Gough
*/
double
gsl_ran_gamma_mt (const gsl_rng * r, const double a, const double b)
{
return gsl_ran_gamma (r, a, b);
}
double
gsl_ran_gamma (const gsl_rng * r, const double a, const double b)
{
/* assume a > 0 */
if (a < 1)
{
double u = gsl_rng_uniform_pos (r);
return gsl_ran_gamma (r, 1.0 + a, b) * pow (u, 1.0 / a);
}
{
double x, v, u;
double d = a - 1.0 / 3.0;
double c = (1.0 / 3.0) / sqrt (d);
while (1)
{
do
{
x = gsl_ran_gaussian_ziggurat (r, 1.0);
v = 1.0 + c * x;
}
while (v <= 0);
v = v * v * v;
u = gsl_rng_uniform_pos (r);
if (u < 1 - 0.0331 * x * x * x * x)
break;
if (log (u) < 0.5 * x * x + d * (1 - v + log (v)))
break;
}
return b * d * v;
}
}
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