Numerical Verification of Analytical Results for Statistical Description of Polymer Chains in Nematic Systems

Walasek, Janusz; Jedynak, Radosław

doi:10.1002/mats.201400095

Cited by 2 publications

(1 citation statement)

References 21 publications

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“…Its inverse function appears in diverse contexts, such as polymer science [1][2][3][4][5], magnetism [6][7][8][9] and the theory of rubber elasticity [10][11][12][13][14]. Its form makes it impossible to solve explicitly for the argument of the function.…”

Section: Introductionmentioning

confidence: 99%

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

Jedynak

2018

Mathematics and Mechanics of Solids

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This paper is motivated by Marchi and Arruda (Math Mech Solids, 2018), who developed numerous approximations to the inverse Langevin function, minimizing their maximum relative error with the help of special software. We are convinced that this method, together with the minimax approximation, is the most robust and versatile approximation algorithm. It uses one of the first objective functions developed by differential evolution. It guarantees the best solution in one trial. In previous papers, a constraint on the error function of the inverse Langevin function was imposed, but here we discuss the approximants resulting from various types of constraint. The evolution of the different mathematical methods that have been used to approximate the inverse Langevin function will also be treated, providing a useful starting point for researchers beginning to work in the field of approximation theory. To categorize the existing solutions, we find the optimal approximations of the inverse Langevin function for a given complexity and for different constraints on the error function.

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