A comprehensive study of the mathematical methods used to approximate the inverse Langevin function

This paper is motivated by Marchi and Arruda (Math Mech Solids, 2018), who developed numerous approximations to the inverse Langevin function, minimizing their maximum relative error with the help of special software. We are convinced that this method, together with the minimax approximation, is the most robust and versatile approximation algorithm. It uses one of the first objective functions developed by differential evolution. It guarantees the best solution in one trial. In previous papers, a constraint on the error function of the inverse Langevin function was imposed, but here we discuss the approximants resulting from various types of constraint. The evolution of the different mathematical methods that have been used to approximate the inverse Langevin function will also be treated, providing a useful starting point for researchers beginning to work in the field of approximation theory. To categorize the existing solutions, we find the optimal approximations of the inverse Langevin function for a given complexity and for different constraints on the error function.