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/* randist/gauss.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2006 James Theiler, Brian Gough
* Copyright (C) 2006 Charles Karney
*
* 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_rng.h>
#include <gsl/gsl_randist.h>
/* Of the two methods provided below, I think the Polar method is more
* efficient, but only when you are actually producing two random
* deviates. We don't produce two, because then we'd have to save one
* in a static variable for the next call, and that would screws up
* re-entrant or threaded code, so we only produce one. This makes
* the Ratio method suddenly more appealing.
*
* [Added by Charles Karney] We use Leva's implementation of the Ratio
* method which avoids calling log() nearly all the time and makes the
* Ratio method faster than the Polar method (when it produces just one
* result per call). Timing per call (gcc -O2 on 866MHz Pentium,
* average over 10^8 calls)
*
* Polar method: 660 ns
* Ratio method: 368 ns
*
*/
/* Polar (Box-Mueller) method; See Knuth v2, 3rd ed, p122 */
double
gsl_ran_gaussian (const gsl_rng * r, const double sigma)
{
double x, y, r2;
do
{
/* choose x,y in uniform square (-1,-1) to (+1,+1) */
x = -1 + 2 * gsl_rng_uniform_pos (r);
y = -1 + 2 * gsl_rng_uniform_pos (r);
/* see if it is in the unit circle */
r2 = x * x + y * y;
}
while (r2 > 1.0 || r2 == 0);
/* Box-Muller transform */
return sigma * y * sqrt (-2.0 * log (r2) / r2);
}
/* Ratio method (Kinderman-Monahan); see Knuth v2, 3rd ed, p130.
* K+M, ACM Trans Math Software 3 (1977) 257-260.
*
* [Added by Charles Karney] This is an implementation of Leva's
* modifications to the original K+M method; see:
* J. L. Leva, ACM Trans Math Software 18 (1992) 449-453 and 454-455. */
double
gsl_ran_gaussian_ratio_method (const gsl_rng * r, const double sigma)
{
double u, v, x, y, Q;
const double s = 0.449871; /* Constants from Leva */
const double t = -0.386595;
const double a = 0.19600;
const double b = 0.25472;
const double r1 = 0.27597;
const double r2 = 0.27846;
do /* This loop is executed 1.369 times on average */
{
/* Generate a point P = (u, v) uniform in a rectangle enclosing
the K+M region v^2 <= - 4 u^2 log(u). */
/* u in (0, 1] to avoid singularity at u = 0 */
u = 1 - gsl_rng_uniform (r);
/* v is in the asymmetric interval [-0.5, 0.5). However v = -0.5
is rejected in the last part of the while clause. The
resulting normal deviate is strictly symmetric about 0
(provided that v is symmetric once v = -0.5 is excluded). */
v = gsl_rng_uniform (r) - 0.5;
/* Constant 1.7156 > sqrt(8/e) (for accuracy); but not by too
much (for efficiency). */
v *= 1.7156;
/* Compute Leva's quadratic form Q */
x = u - s;
y = fabs (v) - t;
Q = x * x + y * (a * y - b * x);
/* Accept P if Q < r1 (Leva) */
/* Reject P if Q > r2 (Leva) */
/* Accept if v^2 <= -4 u^2 log(u) (K+M) */
/* This final test is executed 0.012 times on average. */
}
while (Q >= r1 && (Q > r2 || v * v > -4 * u * u * log (u)));
return sigma * (v / u); /* Return slope */
}
double
gsl_ran_gaussian_pdf (const double x, const double sigma)
{
double u = x / fabs (sigma);
double p = (1 / (sqrt (2 * M_PI) * fabs (sigma))) * exp (-u * u / 2);
return p;
}
double
gsl_ran_ugaussian (const gsl_rng * r)
{
return gsl_ran_gaussian (r, 1.0);
}
double
gsl_ran_ugaussian_ratio_method (const gsl_rng * r)
{
return gsl_ran_gaussian_ratio_method (r, 1.0);
}
double
gsl_ran_ugaussian_pdf (const double x)
{
return gsl_ran_gaussian_pdf (x, 1.0);
}
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