Analytical Approximations for the Principal Branch of the Lambert W Function

Abstract

A geometric based approach for specifying approximations to the Lambert W function, which can achieve any set relative error bound over the interval [0, ∞), is detailed. Approximations that can achieve arbitrarily high accuracy for the interval [-1/e, 0], based on a two point spline approximation, are specified. Iterative methods can be used to improve the accuracy of the approximations.
Applications include, first, analytical expressions, with set relative error bounds, for the Lambert W function over the interval [0, ∞). Second, approximations, with an arbitrarily low relative error, for upper and lower bounds for the Lambert W function. Third, analytical expressions for the evaluation of and the integral of ⌊W(y)⌋, for y∈[0, ∞), without knowledge of W(y). Fourth, a direct approach for evaluating the Lambert W function to achieve a prior set error constraint.