Analytical approximations for the inverse Langevin function via linearization, error approximation, and iteration

Abstract

This paper details an analytical framework, based on an intermediate function, which facilitates analytical approximations for the inverse Langevin function—a function without an explicit analytical form. The approximations have relative error bounds that are typically much lower than those reported in the literature and which can be made arbitrarily small. Results include convergent series expansions in terms of polynomials and sinusoids which have modest relative error bounds and convergence properties but are convergent over the domain of the inverse Langevin function. An important advance is to use error approximations, and then iterative relationships, which allow simple initial approximations for the inverse Langevin function, with modest relative errors, to generate approximations with arbitrarily low relative errors. One example is that of an initial approximating function, with a relative error bound of 0.00969, which yields relative error bounds of 2.77 × 10−6 and 2.66 × 10−16 after the use of first-order error approximation and then first-order iteration. Functions with much lower error bounds are possible and are detailed. First- and second-order Taylor series can be used to simplify the error- and iteration-based approximations.

Scaling relations for brittle fracture of entangled polystyrene melts and solutions in elongational flow

“…We also tested second-and third-order functions presented recently by Howard [41] which are close to optimum in the sense of minimizing the magnitude of the maximum relative error in the approximation of the inverse Langevin function and found no significant difference to the use of Eq. ( 23).…”

Schröder-Based Inverse Function Approximation

“…and are defined, respectively, in [20][21][22]. Their respective relative error bounds, associated with the interval [0, 1) , are: 9.69 × 10 −3 , 1.79 × 10 −3 and 7.22 × 10 −4 .…”

“…Their respective relative error bounds, associated with the interval [0, 1) , are: 9.69 × 10 −3 , 1.79 × 10 −3 and 7.22 × 10 −4 . The papers [1,2,20,23,24], for example, detail alternative approximations.…”

“…An alternative, but less well known, approach for approximating the root of a function f is to directly utilize the inverse function f −1 , with the result being Schröder's approximations of the first kind. Petković [11] (Equation ( 17)), Gdawiec [3] (Equation (20)) and Dubeau [11] (Section 3) provide a perspective, and the original paper by Schröder dates from 1870 [12] (Equation (21). The focus of this paper is on utilizing Schröder's approximations of the first kind, modified for the inverse function approximation case, to establish general analytical approximation forms for an inverse function whose explicit analytical form is not known.…”

Relationship between Inverse Langevin Function and r0-r1-Lambert W Function

“…Whilst the Langevin function is well defined, its inverse, denoted L −1 , does not have a known analytical form and significant research efforts have led to many approximations with the papers, for example, and in chronological order, by Itskov (2012) [4], Nguessong (2014) [5], Darabi (2015) [6], Jedynak (2015) [7], Kröger (2015) [8], Marchi (2015) [9], Rickaby (2015) [10], Petrosyan (2017) [11], Jedynak (2018) [12], Marchi (2019) [13], and Howard (2020) [14], providing useful approximations. Graphs of the Langevin function and the inverse Langevin function are shown in Figure 1.…”