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.