The stability of extended Floater-Hormann interpolants

TL;DR

This paper analyzes the stability of extended Floater-Hormann interpolants, revealing exponential growth of the Lebesgue constant and identifying sources of numerical instability, especially in the extrapolation step.

Contribution

It provides a new stability analysis showing the Lebesgue constant can grow exponentially and highlights the instability caused by the extrapolation step in extended interpolants.

Findings

Lebesgue constant can grow exponentially with parameters

The barycentric formula used is often backward unstable

Extrapolation significantly contributes to numerical instability

Abstract

We present a new analysis of the stability of extended Floater-Hormann interpolants, in which both noisy data and rounding errors are considered. Contrary to what is claimed in the current literature, we show that the Lebesgue constant of these interpolants can grow exponentially with the parameters that define them, and we emphasize the importance of using the proper interpretation of the Lebesgue constant in order to estimate correctly the effects of noise and rounding errors. We also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants. Our experiments show that extended interpolants mentioned in the literature satisfy this condition and, therefore, the formula used to implement them is not backward stable. Finally, we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation.