The stability of extended Floater-Hormann interpolants
For numerical analysts using extended Floater-Hormann interpolants, this work corrects misconceptions about stability and highlights practical numerical issues.
The paper shows that the Lebesgue constant of extended Floater-Hormann interpolants can grow exponentially with parameters, contradicting prior claims, and that the barycentric formula used for implementation is not backward stable. Experiments confirm instability for interpolants in the literature.
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.