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Numerische Mathematik

, Volume 140, Issue 4, pp 939–962 | Cite as

An iterative approach to barycentric rational Hermite interpolation

  • Emiliano Cirillo
  • Kai HormannEmail author
Article
  • 175 Downloads

Abstract

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at \(n+1\) nodes and then successively adds m correction terms to fit the data up to the mth derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater–Hormann interpolants, which are based on blending local polynomial interpolants of degree d with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most \((m+1)(n+1)-1\) and \((m+1)(n-d)\), respectively. They converge at the rate of \(O(h^{(m+1)(d+1)})\) as the mesh size h converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for \(m=1\) and \(m=2\), and show that the approximation results compare favourably with other constructions.

Mathematics Subject Classification

65D05 41A05 41A20 41A25 

Notes

Acknowledgements

This work was supported by the Swiss National Science Foundation (SNSF) under Project Number 200021_150053. We thank the anonymous reviewers for their insightful comments, which helped to improve this paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of InformaticsUniversità della Svizzera italianaLuganoSwitzerland

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