On Fast Implementation of Higher Order Hermite-Fejer Interpolation

TL;DR

This paper presents a fast, stable method for higher order Hermite-Fejer interpolation at Gauss-Jacobi points, reducing computational complexity to linear operations and improving numerical stability.

Contribution

It introduces a new efficient implementation based on Sturm-Liouville equations that cancels exponential factors, enabling linear-time computation of barycentric weights for Hermite-Fejer interpolation.

Findings

Barycentric weights can be evaluated efficiently with linear complexity.

The method cancels exponential growth factors, enhancing stability.

Convergence rates are established for interpolation at Gauss-Jacobi points.

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.