Analytical approximations for real values of the Lambert W-function

摘要

The Lambert W is a transcendental function defined by solutions of the equation W exp(W) = x. For real values of the argument, x, the W-function has two branches, W-0 (the principal branch) and W-1 (the negative branch). A survey of the literature reveals that, in the case of the principal branch (W-0), the vast majority of W-function applications use, at any given time, only a portion of the branch viz. the parts defined by the ranges -1 less than or equal to W-0 less than or equal to 0 and 0 less than or equal to W-0. Approximations are presented for each portion of W-0, and for W-1. It is shown that the present approximations are very accurate with relative errors down to around 0.02 or smaller. The approximations can be used directly, or as starting values for iterative improvement schemes. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved. References: 41