On the numerical stability of linear barycentric rational interpolation
Additional information
Authors
Fuda C.,
Campagna R.,
Hormann K.
Type
Journal Article
Year
2022
Language
English
Abstract
The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as ๐(๐ฅ)=โ๐๐=0๐๐(๐ฅ)๐๐/โ๐๐=0๐๐(๐ฅ) in terms of data values ๐๐ and some functions ๐๐ and ๐๐ for ๐=0,โฆ,๐ and ๐=0,โฆ,๐ with a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows with n, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless of n. We support our theoretical results with numerical experiments.
Journal
Numerische Mathematik
Volume
152
Number ( Month )
4
Pages (or article number)
761 - 786
Diffusion
License
CC BY
Visibility
Public
Status open access
Hybrid
