--- /dev/null
+%% Copyright (C) 1998 by Nicol N. Schraudolph
+%%
+%% This program is free software; you can redistribute and/or
+%% modify it under the terms of the GNU General Public
+%% License as published by the Free Software Foundation;
+%% either version 3, 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 software; see the file COPYING. If not,
+%% see <http://www.gnu.org/licenses/>.
+
+## -*- texinfo -*-
+## @deftypefn {Function File} {@var{x} = } lambertw (@var{z})
+## @deftypefnx {Function File} {@var{x} = } lambertw (@var{z}, @var{n})
+## Compute the Lambert W function of @var{z}.
+##
+## This function satisfies W(z).*exp(W(z)) = z, and can thus be used to express
+## solutions of transcendental equations involving exponentials or logarithms.
+##
+## @var{n} must be integer, and specifies the branch of W to be computed;
+## W(z) is a shorthand for W(0,z), the principal branch. Branches
+## 0 and -1 are the only ones that can take on non-complex values.
+##
+## If either @var{n} or @var{z} are non-scalar, the function is mapped to each
+## element; both may be non-scalar provided their dimensions agree.
+##
+## This implementation should return values within 2.5*eps of its
+## counterpart in Maple V, release 3 or later. Please report any
+## discrepancies to the author, Nici Schraudolph <schraudo@@inf.ethz.ch>.
+##
+## For further details, see:
+##
+## Corless, Gonnet, Hare, Jeffrey, and Knuth (1996), `On the Lambert
+## W Function', Advances in Computational Mathematics 5(4):329-359.
+## @end deftypefn
+
+%% Author: Nicol N. Schraudolph <schraudo@inf.ethz.ch>
+%% Version: 1.0
+%% Created: 07 Aug 1998
+%% Keywords: Lambert W Omega special transcendental function
+
+function w = lambertw(b,z)
+ if (nargin == 1)
+ z = b;
+ b = 0;
+ else
+ %% some error checking
+ if (nargin != 2)
+ print_usage;
+ else
+ if (any(round(real(b)) != b))
+ usage('branch number for lambertw must be integer')
+ end
+ end
+ end
+
+ %% series expansion about -1/e
+ %
+ % p = (1 - 2*abs(b)).*sqrt(2*e*z + 2);
+ % w = (11/72)*p;
+ % w = (w - 1/3).*p;
+ % w = (w + 1).*p - 1
+ %
+ % first-order version suffices:
+ %
+ w = (1 - 2*abs(b)).*sqrt(2*e*z + 2) - 1;
+
+ %% asymptotic expansion at 0 and Inf
+ %
+ v = log(z + ~(z | b)) + 2*pi*I*b;
+ v = v - log(v + ~v);
+
+ %% choose strategy for initial guess
+ %
+ c = abs(z + 1/e);
+ c = (c > 1.45 - 1.1*abs(b));
+ c = c | (b.*imag(z) > 0) | (~imag(z) & (b == 1));
+ w = (1 - c).*w + c.*v;
+
+ %% Halley iteration
+ %
+ for n = 1:10
+ p = exp(w);
+ t = w.*p - z;
+ f = (w != -1);
+ t = f.*t./(p.*(w + f) - 0.5*(w + 2.0).*t./(w + f));
+ w = w - t;
+ if (abs(real(t)) < (2.48*eps)*(1.0 + abs(real(w)))
+ && abs(imag(t)) < (2.48*eps)*(1.0 + abs(imag(w))))
+ return
+ end
+ end
+ warning('iteration limit reached, result of lambertw may be inaccurate');
+end
+