| /* 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)); |
| } |