Generalized error-minimizing, rational inverse Langevin approximations

Abstract

The Langevin function is a non-invertible function, whose inverse commonly appears in statistical mechanics problems, particularly network models of rubber elasticity. This non-invertibility often results in the use of approximations. Owing to the prevalence of the inverse Langevin function, numerous forms of approximate have been proposed. Rational approximates are often employed because of their ability to admit asymptotic behavior in finite domains, similar to the exact inverse Langevin function. Despite the desired asymptotics of rational approximates, there is unavoidable error associated with the approximate within its domain. In this work, an error-minimizing approach for determining specific forms of rational approximates is generalized to approximates of arbitrary numerator and denominator orders. By expanding to general orders of rational approximates, the best approximate can be selected for an application based on either maximum relative error or function form considerations.

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

“…The general solution is given by equation (27). Similar forms had been motivated earlier in the literature (equation (8) in Kro¨ger [1] and equation ( 12) in Itskov et al [44]).…”

“…In our sense, this idea is identical to our proposal of using both constraints on the relative error function at its ends. The approximants provided by Marchi and Arruda [40,44] follow this rule.…”

“…Now we are ready to introduce the new method of solving the minimax approximation. First, we recall the objective function which was used by Marchi and Arruda [40,44].…”

“…Marchi and Arruda [44] reported that f MA obj had to be evaluated at 10,000 points at each iteration of the optimization routine. This formula does not directly reflect the characteristic equi-oscillation property of the best solution; therefore, we can expect that the result after one trial of optimization can be near to the best approximant.…”

“…Marchi and Arruda [44] extensively studied optimal approximations in the form of rational functions, for orders of the numerator and denominator ranging from 1 to 7.…”