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- /* specfunc/lambert.c
- *
- * Copyright (C) 2007 Brian Gough
- * 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 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.
- */
- /* Author: G. Jungman */
- #include "gsl__config.h"
- #include <math.h>
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "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 = 10 * 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 "gsl_specfunc__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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