Guaranteed- and high-precision evaluation of the Lambert [formula omitted] function

Abstract

Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert W function. The W function, also occurring frequently in many branches of science, is a non-elementary but now standard mathematical function implemented in all major technical computing systems. In this work, we analyze an efficient logarithmic recursion with quadratic convergence rate to approximate its two real branches, W0 and W−1. We propose suitable starting values that ensure monotone convergence on the whole domain of definition of both branches. Then, we provide a priori, simple, explicit and uniform estimates on the convergence speed, which enable guaranteed, high-precision approximations of W0 and W−1 at any point.