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