Journal article

On the numerical stability of linear barycentric rational interpolation

  • Fuda, Chiara Facoltà di scienze informatiche, Università della Svizzera italiana, Svizzera
  • Campagna, Rosanna ORCID Dipartimento di Matematica e Fisica, Università degli studi della Campania, Caserta, Italy
  • Hormann , Kai ORCID Facoltà di scienze informatiche, Università della Svizzera italiana, Svizzera
  • 2022
Published in:
  • Numerische Mathematik. - 2022, vol. 152, no. 4, p. 761 - 786
English 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 𝑟(𝑥)=∑𝑛𝑖=0𝑎𝑖(𝑥)𝑓𝑖/∑𝑚𝑗=0𝑏𝑗(𝑥) in terms of data values 𝑓𝑖 and some functions 𝑎𝑖 and 𝑏𝑗 for 𝑖=0,…,𝑛 and 𝑗=0,…,𝑚 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.
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