A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

TL;DR

This paper introduces a new numerical method for computing the inverse Langevin function with near machine precision accuracy, combining discretization, rational functions, and spline interpolation for high efficiency and accuracy.

Contribution

A simple, efficient numerical procedure that achieves near machine precision accuracy for the inverse Langevin function across its domain, outperforming existing approximants.

Findings

Achieves near machine precision accuracy in inverse Langevin function calculations.

Maintains computational efficiency comparable to the fastest existing methods.

Provides Matlab code for implementation and comparison.

Abstract

The inverse Langevin function is a fundamental part of the statistical chain models used to describe the behavior of polymeric-like materials, appearing also in other fields such as magnetism, molecular dynamics and even biomechanics. In the last four years, several approximants of the inverse Langevin function have been proposed. In most of them, optimization techniques are used to reduce the relative error of previously published approximants to reach orders of magnitude of O(10^-2, 10^-3). In this paper a new simple and efficient numerical approach to calculate the inverse Langevin function is proposed. Its main feature is the reduction of the relative errors in all the domain x=[0,1) to near machine precision, maintaining function evaluation CPU times similar to those of the most efficient approximants. The method consists in the discretization of the Langevin function, the calculation of the inverse of these discretization points and their interpolation by cubic splines. In order to reproduce the asymptotic behavior of the inverse Langevin function, a rational function is considered only in the asymptotic zone keeping C^1 continuity with the cubic splines. We include customizable Matlab codes to create the spline coefficients, to evaluate the function, and to compare accuracy and efficiency with other published proposals.