Global approximations to the principal real-valued branch of the Lambert W-function Original Research Article

W(z) is defined implicitly as the root of W exp(W) = z. It is shown that a simple analytic approximation has a relative error of less than 5% over the whole domain zϵ [−exp(−1), ∞] of the principle branch—sufficiently accurate so that four Newton iterations refine this approximation to a relative error smaller than 1.E-12. As a second form of global approximation, the W-function is expanded as a series of rational Chebyshev functions TBj in a shifted, logarithmic coordinate with an error that decreases exponentially fast with the series truncation.