There have been quite a few posts recently regarding problems whose
solutions involve the Lambert W-function.  I have an iterative algorithm
that I use for computing this function that converges fairly quickly.
First, let's look at the inverse of W(x), w e^w.

Analysis of x e^x
For w > 0, w e^w increases monotonically from 0 to infinity.  When w is
negative, w e^w is negative.  Thus, for x > 0, W(x) is positive and well
defined and increases monotonically.

For w < 0, w e^w reaches a minimum of -1/e at -1.  On (-1,0), w e^w
increases monotonically from -1/e to 0.  On (-oo,-1), w e^w decreases
monotonically from 0 to -1/e.  Thus, on (-1/e,0), W(x) can have one of
two values, one in (-1,0) and another in (-oo,-1).  The value in (-1,0)
is called the principal value.

The iteration
Using Newton's method for solving w e^w yields the following iterative
step for finding W(x):

           x e^-w + w^2
    w    = ------------                                              [1]
     new       w + 1

Initial values of w
For the principal value when -1/e <= x < 0 and when 0 <= x <= 10, use
w = 0.  When x > 10, use w = log(x) - log(log(x)).

For the non-principal value, if x is in [-1/e,-.1], use w = -2; and
if x is in (-.1,0), use w = log(-x) - log(-log(-x)).

Another iteration
When x is near -1/e, [1] converges a little slowly.  I use a parabolic
iteration method in those cases.

                              x + 1/e
    w    = -1 + (w+1) sqrt( ----------- )                            [2]
     new                    w e^w + 1/e

Using an initial w of 0 for the principal value, and w = -2 for the
non-principal value.

Rob Johnson