On the linearization methods for univariate Birkhoff rational interpolation

Abstract

As a natural extension of Birkhoff polynomial interpolation, Birkhoff rational interpolation is difficult to be linearized. In this work, we strategically split the univariate Birkhoff rational interpolation into multiple subproblems, such that these subproblems can be linearized. Due to the interpolating function may not be unique, we innovatively introduce a special condition such that a recurrence solution formula can be constructed if the condition is satisfied. If we omit the special condition, we also propose a method to obtain the rational interpolating function through solving a linear system. The later method tends to give a lower degree interpolating function with better approximation accuracy and while the former tends to provide less computing cost. Experiments show the efficacies of these rational interpolating methods and indicate potential benefits of these methods over the polynomial interpolation method.