Guaranteed- and high-precision evaluation of the Lambert W function

https://doi.org/10.1016/j.amc.2022.127406Get rights and content
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Highlights

  • Recursive approximations of both real branches of the Lambert W function via elementary functions.

  • Monotone recursive approximations with explicit starting values and simple, uniform error estimates.

  • Recursive approximations with quadratic convergence rate.

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 W1. 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 W1 at any point.

Keywords

Lambert W function
Explicit estimates
Recursive approximations

Data Availability

  • No data was used for the research described in the article.

Cited by (0)

The project “Application-domain specific highly reliable IT solutions” has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme TKP2020-NKA-06 (National Challenges Subprogramme) funding scheme.