Computing the Lambert W function in arbitrary-precision complex interval arithmetic

TL;DR

This paper presents a rigorous algorithm for computing all branches of the Lambert W function in complex interval arithmetic, implemented in the Arb library, enhancing precision management and branch cut handling.

Contribution

It introduces a new algorithm for evaluating the Lambert W function with rigorous error bounds in complex interval arithmetic, improving upon previous heuristic methods.

Findings

Algorithm provides rigorous error bounds for all branches

Implementation in Arb library enables practical computations

Enhances precision control and branch cut management

Abstract

We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in interval arithmetic, which has been implemented in the Arb library. The classic 1996 paper on the Lambert W function by Corless et al. provides a thorough but partly heuristic numerical analysis which needs to be complemented with some explicit inequalities and practical observations about managing precision and branch cuts.