Analytical Approximations for the Principal Branch of the Lambert W Function

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.