A one-pass predictor-corrector algorithm for the inverse Langevin function

Abstract

Inverse Langevin function has an extensive use in statistical mechanics, polymer chemistry, and physics. Main struggle is that the inverse Langevin function cannot be expressed in an exact analytical form. To this end, many approaches to estimate the inverse Langevin function have been proposed. A trade-off can be observed between level of accuracy and mathematical complexity in the existing approximants in the literature. In the present contribution, a simple, yet efficient one-pass predictor-corrector algorithm is proposed for the accurate prediction of the inverse Langevin function. The predictor step uses the approximants y_p(x) proposed in the literature. The corrector term is based on a single iteration applied to the error function $\mathcal{L}$ (y_p) – x. The correction term is based on the y_p and has the same form irrespective of the specific approximant used in the predictor step. Hence, the additional computational cost is constant irrespective of the approximant used as predictor. In order to demonstrate the accuracy and efficiency of the approach, maximum relative error and the computational cost before and after corrector step are analysed for eight different approximants. The proposed one-pass predictor-corrector approach allows the use of relatively simpler approximants of the inverse Langevin function by improving their accuracy by at least an order of magnitude.