Optimal stability of the Lagrange formula and conditioning of the Newton formula
type: Item
instance of: scholarly article
title: Optimal stability of the Lagrange formula and conditioning of the Newton formula
zbMATH Open document ID: 1407.41002
author: J. M. Carnicer
author: Yasmina Khiar
author: J. M. Peña
published in: Journal of Approximation Theory
publication date: 2019-01-18T00:00:00Z
full work available at URL: http://zaguan.unizar.es/record/78093
review text: In this paper, both Lagrange and Newton interpolation formulae are considered with respect to conditioning and stability. A special, i.e., pointwise, condition number is introduced for this interpolation problem, and it is shown that with respect to this particular condition number, the Lagrange expression for the polynomial interpolation is the best. Next, there is a relation of the stability of the Newton description of the polynomial interpolant in comparison to the Lagrange formulation (where a so-called Leja ordering of the interpolation nodes can improve the situation which is otherwise known to be critical). The behaviour that can be observed with Newton as compared to Lagrange can even be identified at (at least) one of the nodes for which the pointwise condition number is useful. In fact, both conditioning and the mentioned stability are regarded in this article. The comparison of Newton and Lagrange is considered in the paper not only for ordinary nodes but specifically also for Leja nodes. In fact, the best uniform upper bounds for conditioning of the Newton situation are very concretely \(3^n\) while we have a bounds of \(2^n\) for the Lagrange case. These are obtained by using \(n+1\) equally spaced interpolation points. The bounds are attained at the final node.
reviewed by: Martin D. Buhmann
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 65D05
zbMATH DE Number: 7003504
zbMATH Keywords: Skeel condition number
zbMATH Keywords: Lagrange interpolation formula
zbMATH Keywords: Newton interpolation formula
zbMATH Keywords: Leja order
describes a project that uses: mctoolbox
MaRDI profile type: Publication
OpenAlex ID: W2736010211
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DOI: 10.1016/J.JAT.2017.07.005
description: scientific article; zbMATH DE number 7003504