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qnorm.c
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qnorm.c
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/*
* Mathlib : A C Library of Special Functions
* Copyright (C) 2000--2020 The R Core Team
* Copyright (C) 1998 Ross Ihaka
* based on AS 111 (C) 1977 Royal Statistical Society
* and on AS 241 (C) 1988 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, a copy is available at
* https://www.R-project.org/Licenses/
*
* SYNOPSIS
*
* double qnorm5(double p, double mu, double sigma,
* int lower_tail, int log_p)
* {qnorm (..) is synonymous and preferred inside R}
*
* DESCRIPTION
*
* Compute the quantile function for the normal distribution.
*
* For small to moderate probabilities, algorithm referenced
* below is used to obtain an initial approximation which is
* polished with a final Newton step.
*
* For very large arguments, an algorithm of Wichura is used.
*
* REFERENCE
*
* Beasley, J. D. and S. G. Springer (1977).
* Algorithm AS 111: The percentage points of the normal distribution,
* Applied Statistics, 26, 118-121.
*
* Wichura, M.J. (1988).
* Algorithm AS 241: The Percentage Points of the Normal Distribution.
* Applied Statistics, 37, 477-484.
*/
#include "nmath.h"
#include "dpq.h"
double qnorm5(double p, double mu, double sigma, int lower_tail, int log_p)
{
double p_, q, r, val;
#ifdef IEEE_754
if (ISNAN(p) || ISNAN(mu) || ISNAN(sigma))
return p + mu + sigma;
#endif
R_Q_P01_boundaries(p, ML_NEGINF, ML_POSINF);
if(sigma < 0) ML_WARN_return_NAN;
if(sigma == 0) return mu;
p_ = R_DT_qIv(p);/* real lower_tail prob. p */
q = p_ - 0.5;
#ifdef DEBUG_qnorm
REprintf("qnorm(p=%10.7g, m=%g, s=%g, l.t.= %d, log= %d): q = %g\n",
p,mu,sigma, lower_tail, log_p, q);
#endif
/*-- use AS 241 --- */
/* double ppnd16_(double *p, long *ifault)*/
/* ALGORITHM AS241 APPL. STATIST. (1988) VOL. 37, NO. 3
Produces the normal deviate Z corresponding to a given lower
tail area of P; Z is accurate to about 1 part in 10**16.
(original fortran code used PARAMETER(..) for the coefficients
and provided hash codes for checking them...)
*/
if (fabs(q) <= .425) {/* |p~ - 0.5| <= .425 <==> 0.075 <= p~ <= 0.925 */
r = .180625 - q * q; // = .425^2 - q^2 >= 0
val =
q * (((((((r * 2509.0809287301226727 +
33430.575583588128105) * r + 67265.770927008700853) * r +
45921.953931549871457) * r + 13731.693765509461125) * r +
1971.5909503065514427) * r + 133.14166789178437745) * r +
3.387132872796366608)
/ (((((((r * 5226.495278852854561 +
28729.085735721942674) * r + 39307.89580009271061) * r +
21213.794301586595867) * r + 5394.1960214247511077) * r +
687.1870074920579083) * r + 42.313330701600911252) * r + 1.);
}
else { /* closer than 0.075 from {0,1} boundary :
* r := log(p~); p~ = min(p, 1-p) < 0.075 : */
if(log_p && ((lower_tail && q <= 0) || (!lower_tail && q > 0))) {
r = p;
} else {
r = log( (q > 0) ? R_DT_CIv(p) /* 1-p */ : p_ /* = R_DT_Iv(p) ^= p */);
}
// r = sqrt( - log(min(p,1-p)) ) <==> min(p, 1-p) = exp( - r^2 ) :
r = sqrt(-r);
#ifdef DEBUG_qnorm
REprintf("\t close to 0 or 1: r = %7g\n", r);
#endif
if (r <= 5.) { /* <==> min(p,1-p) >= exp(-25) ~= 1.3888e-11 */
r += -1.6;
val = (((((((r * 7.7454501427834140764e-4 +
.0227238449892691845833) * r + .24178072517745061177) *
r + 1.27045825245236838258) * r +
3.64784832476320460504) * r + 5.7694972214606914055) *
r + 4.6303378461565452959) * r +
1.42343711074968357734)
/ (((((((r *
1.05075007164441684324e-9 + 5.475938084995344946e-4) *
r + .0151986665636164571966) * r +
.14810397642748007459) * r + .68976733498510000455) *
r + 1.6763848301838038494) * r +
2.05319162663775882187) * r + 1.);
}
else if(r >= 816) { // p is *extremly* close to 0 or 1 - only possibly when log_p =TRUE
// Using the asymptotical formula -- is *not* optimal but uniformly better than branch below
val = r * M_SQRT2;
}
else { // p is very close to 0 or 1: r > 5 <==> min(p,1-p) < exp(-25) = 1.3888..e-11
// Wichura, p.478: minimax rational approx R_3(t) is for 5 <= t <= 27 (t :== r)
r += -5.;
val = (((((((r * 2.01033439929228813265e-7 +
2.71155556874348757815e-5) * r +
.0012426609473880784386) * r + .026532189526576123093) *
r + .29656057182850489123) * r +
1.7848265399172913358) * r + 5.4637849111641143699) *
r + 6.6579046435011037772)
/ (((((((r *
2.04426310338993978564e-15 + 1.4215117583164458887e-7)*
r + 1.8463183175100546818e-5) * r +
7.868691311456132591e-4) * r + .0148753612908506148525)
* r + .13692988092273580531) * r +
.59983220655588793769) * r + 1.);
}
if(q < 0.0)
val = -val;
/* return (q >= 0.)? r : -r ;*/
}
return mu + sigma * val;
}