A generalized approach to generate optimized approximations of the inverse Langevin function

Proper approximation of the inverse Langevin function (ILF) is a well-recognized problem with significant relevance in many fields ranging from polymer physics to turbulence mechanics. While several estimations of the ILF has been developed recently, the accuracy/complexity trade-off has remained a major challenge in ILF estimation. Accurate estimations are computationally too expensive, and the low-cost estimations lack high accuracy. Here, a novel approach is developed that can provide a family of approximation functions for ILF with different degrees of accuracy. In the present paper, a simple procedure is presented, which can take current approximation functions with asymptotic behavior and enhance them by the addition of a power series of their induced error. The total error is thus correlated with a number of terms in the power series. We further propose different approaches to reduce the terms of the power series and increase the accuracy, the proposed approach is applied to four different classes of ILF approximations and shows significant improvement. The accuracy/complexity trade-off for the family of ILF approximations generated by the proposed approach is compared against those of other approaches to show the advantage of the proposed model. The level of error of this method can reach as low as 0.02%