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/* cdf/inverse_normal.c
*
* Copyright (C) 2002 Przemyslaw Sliwa and Jason H. Stover.
*
* 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 3 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.
*/
/*
* Computes the inverse normal cumulative distribution function
* according to the algorithm shown in
*
* Wichura, M.J. (1988).
* Algorithm AS 241: The Percentage Points of the Normal Distribution.
* Applied Statistics, 37, 477-484.
*/
#include <config.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_cdf.h>
#include "rat_eval.h"
static double
small (double q)
{
const double a[8] = { 3.387132872796366608, 133.14166789178437745,
1971.5909503065514427, 13731.693765509461125,
45921.953931549871457, 67265.770927008700853,
33430.575583588128105, 2509.0809287301226727
};
const double b[8] = { 1.0, 42.313330701600911252,
687.1870074920579083, 5394.1960214247511077,
21213.794301586595867, 39307.89580009271061,
28729.085735721942674, 5226.495278852854561
};
double r = 0.180625 - q * q;
double x = q * rat_eval (a, 8, b, 8, r);
return x;
}
static double
intermediate (double r)
{
const double a[] = { 1.42343711074968357734, 4.6303378461565452959,
5.7694972214606914055, 3.64784832476320460504,
1.27045825245236838258, 0.24178072517745061177,
0.0227238449892691845833, 7.7454501427834140764e-4
};
const double b[] = { 1.0, 2.05319162663775882187,
1.6763848301838038494, 0.68976733498510000455,
0.14810397642748007459, 0.0151986665636164571966,
5.475938084995344946e-4, 1.05075007164441684324e-9
};
double x = rat_eval (a, 8, b, 8, (r - 1.6));
return x;
}
static double
tail (double r)
{
const double a[] = { 6.6579046435011037772, 5.4637849111641143699,
1.7848265399172913358, 0.29656057182850489123,
0.026532189526576123093, 0.0012426609473880784386,
2.71155556874348757815e-5, 2.01033439929228813265e-7
};
const double b[] = { 1.0, 0.59983220655588793769,
0.13692988092273580531, 0.0148753612908506148525,
7.868691311456132591e-4, 1.8463183175100546818e-5,
1.4215117583164458887e-7, 2.04426310338993978564e-15
};
double x = rat_eval (a, 8, b, 8, (r - 5.0));
return x;
}
double
gsl_cdf_ugaussian_Pinv (const double P)
{
double r, x, pp;
double dP = P - 0.5;
if (P == 1.0)
{
return GSL_POSINF;
}
else if (P == 0.0)
{
return GSL_NEGINF;
}
if (fabs (dP) <= 0.425)
{
x = small (dP);
return x;
}
pp = (P < 0.5) ? P : 1.0 - P;
r = sqrt (-log (pp));
if (r <= 5.0)
{
x = intermediate (r);
}
else
{
x = tail (r);
}
if (P < 0.5)
{
return -x;
}
else
{
return x;
}
}
double
gsl_cdf_ugaussian_Qinv (const double Q)
{
double r, x, pp;
double dQ = Q - 0.5;
if (Q == 1.0)
{
return GSL_NEGINF;
}
else if (Q == 0.0)
{
return GSL_POSINF;
}
if (fabs (dQ) <= 0.425)
{
x = small (dQ);
return -x;
}
pp = (Q < 0.5) ? Q : 1.0 - Q;
r = sqrt (-log (pp));
if (r <= 5.0)
{
x = intermediate (r);
}
else
{
x = tail (r);
}
if (Q < 0.5)
{
return x;
}
else
{
return -x;
}
}
double
gsl_cdf_gaussian_Pinv (const double P, const double sigma)
{
return sigma * gsl_cdf_ugaussian_Pinv (P);
}
double
gsl_cdf_gaussian_Qinv (const double Q, const double sigma)
{
return sigma * gsl_cdf_ugaussian_Qinv (Q);
}