The stability of extended Floater–Hormann interpolants

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.

Improved conditioning of the Floater--Hormann interpolants

“…Third, Klein [8] proposed extrapolating to points outside of the interval and constructing Floater-Hormann interpolants the extended set of points. Certain published results suggest that these Extended Floater-Hormann interpolants do not suffer the usual drawbacks from high values of d, but others [5] suggest that the extrapolation process is a source of significant instability. Klein explicitly ignored this source of error in his analysis.…”

“…One proposal to improve the conditioning of the Floater-Hormann interpolants extrapolates to points outside of the interval, and uses a Floater-Hormann interpolant on the extended set of points [8]. This does resolve the source of the ill-conditioning identified above, but at the cost of introducing instability by the extrapolation process [5]. There is evidence that even when the extrapolated points are computed in multiple precision arithmetic, the effect of measurement and rounding errors in the initial y i can negate any advantage of this approach [5].…”

“…This does resolve the source of the ill-conditioning identified above, but at the cost of introducing instability by the extrapolation process [5]. There is evidence that even when the extrapolated points are computed in multiple precision arithmetic, the effect of measurement and rounding errors in the initial y i can negate any advantage of this approach [5]. Moreover, the extrapolation process obscures the dependence of the interpolants on the initial points, makes explicit basis functions difficult to construct, and complicates the study of the Lebesgue function.…”

Moore: Interval Arithmetic in C++20

“…The Moore library was implemented to be used in my research, and the experiments reflect this. I present timings related to my current research about the stability of barycentric interpolation [12,13,4]. In this research I look for parameters w 0 , .…”