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/* specfunc/lambert.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001 Gerard Jungman
*
* 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.
*/
/* Author: G. Jungman */
#include <config.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_lambert.h>
/* Started with code donated by K. Briggs; added
* error estimates, GSL foo, and minor tweaks.
* Some Lambert-ology from
* [Corless, Gonnet, Hare, and Jeffrey, "On Lambert's W Function".]
*/
/* Halley iteration (eqn. 5.12, Corless et al) */
static int
halley_iteration(
double x,
double w_initial,
unsigned int max_iters,
gsl_sf_result * result
)
{
double w = w_initial;
unsigned int i;
for(i=0; i<max_iters; i++) {
double tol;
const double e = exp(w);
const double p = w + 1.0;
double t = w*e - x;
/* printf("FOO: %20.16g %20.16g\n", w, t); */
if (w > 0) {
t = (t/p)/e; /* Newton iteration */
} else {
t /= e*p - 0.5*(p + 1.0)*t/p; /* Halley iteration */
};
w -= t;
tol = GSL_DBL_EPSILON * GSL_MAX_DBL(fabs(w), 1.0/(fabs(p)*e));
if(fabs(t) < tol)
{
result->val = w;
result->err = 2.0*tol;
return GSL_SUCCESS;
}
}
/* should never get here */
result->val = w;
result->err = fabs(w);
return GSL_EMAXITER;
}
/* series which appears for q near zero;
* only the argument is different for the different branches
*/
static double
series_eval(double r)
{
static const double c[12] = {
-1.0,
2.331643981597124203363536062168,
-1.812187885639363490240191647568,
1.936631114492359755363277457668,
-2.353551201881614516821543561516,
3.066858901050631912893148922704,
-4.175335600258177138854984177460,
5.858023729874774148815053846119,
-8.401032217523977370984161688514,
12.250753501314460424,
-18.100697012472442755,
27.029044799010561650
};
const double t_8 = c[8] + r*(c[9] + r*(c[10] + r*c[11]));
const double t_5 = c[5] + r*(c[6] + r*(c[7] + r*t_8));
const double t_1 = c[1] + r*(c[2] + r*(c[3] + r*(c[4] + r*t_5)));
return c[0] + r*t_1;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_lambert_W0_e(double x, gsl_sf_result * result)
{
const double one_over_E = 1.0/M_E;
const double q = x + one_over_E;
if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(q < 0.0) {
/* Strictly speaking this is an error. But because of the
* arithmetic operation connecting x and q, I am a little
* lenient in case of some epsilon overshoot. The following
* answer is quite accurate in that case. Anyway, we have
* to return GSL_EDOM.
*/
result->val = -1.0;
result->err = sqrt(-q);
return GSL_EDOM;
}
else if(q == 0.0) {
result->val = -1.0;
result->err = GSL_DBL_EPSILON; /* cannot error is zero, maybe q == 0 by "accident" */
return GSL_SUCCESS;
}
else if(q < 1.0e-03) {
/* series near -1/E in sqrt(q) */
const double r = sqrt(q);
result->val = series_eval(r);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else {
static const unsigned int MAX_ITERS = 10;
double w;
if (x < 1.0) {
/* obtain initial approximation from series near x=0;
* no need for extra care, since the Halley iteration
* converges nicely on this branch
*/
const double p = sqrt(2.0 * M_E * q);
w = -1.0 + p*(1.0 + p*(-1.0/3.0 + p*11.0/72.0));
}
else {
/* obtain initial approximation from rough asymptotic */
w = log(x);
if(x > 3.0) w -= log(w);
}
return halley_iteration(x, w, MAX_ITERS, result);
}
}
int
gsl_sf_lambert_Wm1_e(double x, gsl_sf_result * result)
{
if(x > 0.0) {
return gsl_sf_lambert_W0_e(x, result);
}
else if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
static const unsigned int MAX_ITERS = 32;
const double one_over_E = 1.0/M_E;
const double q = x + one_over_E;
double w;
if (q < 0.0) {
/* As in the W0 branch above, return some reasonable answer anyway. */
result->val = -1.0;
result->err = sqrt(-q);
return GSL_EDOM;
}
if(x < -1.0e-6) {
/* Obtain initial approximation from series about q = 0,
* as long as we're not very close to x = 0.
* Use full series and try to bail out if q is too small,
* since the Halley iteration has bad convergence properties
* in finite arithmetic for q very small, because the
* increment alternates and p is near zero.
*/
const double r = -sqrt(q);
w = series_eval(r);
if(q < 3.0e-3) {
/* this approximation is good enough */
result->val = w;
result->err = 5.0 * GSL_DBL_EPSILON * fabs(w);
return GSL_SUCCESS;
}
}
else {
/* Obtain initial approximation from asymptotic near zero. */
const double L_1 = log(-x);
const double L_2 = log(-L_1);
w = L_1 - L_2 + L_2/L_1;
}
return halley_iteration(x, w, MAX_ITERS, result);
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_lambert_W0(double x)
{
EVAL_RESULT(gsl_sf_lambert_W0_e(x, &result));
}
double gsl_sf_lambert_Wm1(double x)
{
EVAL_RESULT(gsl_sf_lambert_Wm1_e(x, &result));
}
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