On Fast Implementation of Higher Order Hermite-Fejer Interpolation

Abstract: The problem of barycentric Hermite interpolation is highly susceptible to overflows or underflows. In this paper, based on Sturm-Liouville equations for Jacobi orthogonal polynomials, we consider the fast implementation on the second barycentric formula for higher order Hermite-Fej\'{e}r interpolation at Gauss-Jacobi or Jacobi-Gauss-Lobatto pointsystems, where the barycentric weights can be efficiently evaluated and cost linear operations corresponding to the number of grids totally. Furthermore, due to the division of the second barycentric form, the exponentially increasing common factor in the barycentric weights can be canceled, which yields a superiorly stable method for computing the simplified barycentric weights, and leads to a fast implementation of the higher order Hermite-Fej\'{e}r interpolation with linear operations on the number of grids. In addition, the convergence rates are derived for Hermite-Fej\'{e}r interpolation at Gauss-Jacobi pointsystems.