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Diffstat (limited to 'gsl-1.9/specfunc/lambert.c')
| -rw-r--r-- | gsl-1.9/specfunc/lambert.c | 230 |
1 files changed, 230 insertions, 0 deletions
diff --git a/gsl-1.9/specfunc/lambert.c b/gsl-1.9/specfunc/lambert.c new file mode 100644 index 0000000..896ac58 --- /dev/null +++ b/gsl-1.9/specfunc/lambert.c @@ -0,0 +1,230 @@ +/* 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)); +} |