Explicit and recursive estimates of the Lambert W function

TL;DR

This paper develops new bounds and recursive algorithms for accurately approximating the Lambert W function's real branches, enabling high-precision solutions and settling a conjecture about solutions to a specific exponential equation.

Contribution

It introduces improved bounds and recursive methods with guaranteed convergence for the Lambert W function, along with explicit estimates for high-precision computation.

Findings

Derived analytic bounds for W_0 for large arguments.

Analyzed two logarithmic recursions with different convergence rates.

Provided explicit convergence speed estimates for recursive approximations.

Abstract

Solutions to a wide variety of transcendental equations can be expressed in terms of the Lambert $\mathrm{W}$ function. The $\mathrm{W}$ function, occurring frequently in applications, is a non-elementary, but now standard mathematical function implemented in all major technical computing systems. In this work, we discuss some approximations of the two real branches, $\mathrm{W}_0$ and $\mathrm{W}_{-1}$. On the one hand, we present some analytic lower and upper bounds on $\mathrm{W}_0$ for large arguments that improve on some earlier results in the literature. On the other hand, we analyze two logarithmic recursions, one with linear, and the other with quadratic rate of convergence. We propose suitable starting values for the recursion with quadratic rate that ensure convergence on the whole domain of definition of both real branches. We also provide a priori, simple, explicit and uniform estimates on its convergence speed that enable guaranteed, high-precision approximations of $\mathrm{W}_0$ and $\mathrm{W}_{-1}$ at any point. Finally, as an application of the $\mathrm{W}_0$ function, we settle a conjecture about the growth rate of the positive non-trivial solutions to the equation $x^y=y^x$.