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Rounding error analysis of divided differences schemes: Newton’s divided differences; Neville’s algorithm; Richardson extrapolation; Romberg quadrature; etc.

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Abstract

In this note, we extend the analysis of Camargo (J. Comp. Appl. Math 365, 2020) on the numerical stability of classical Newton’s divided differences to a broader a class of divided differences algorithms that includes Neville’s algorithm for the Lagrange interpolation and some of its particular instances as the Richardson extrapolation and Romberg quadrature. We show that these algorithms are backward stable and we bound the overall numerical error in their computation in finite precision. Our analysis solves a subtle question in numerical Lagrange interpolation that passed unnoticed so far. On the one hand, the Richardson extrapolation and Romberg quadrature have been extensively described as divided differences schemes and their connections with Neville’s algorithm were already highlighted in the literature. On the other hand, the unique algorithm for computing Lagrange interpolants for which backward stability can be ensured (so far) for extrapolation is the first barycentric formula. By showing that Neville’s algorithm is also backward stable for extrapolation, our result consolidates a solid background for the usual representation of the Richardson extrapolation and Romberg quadrature as divided differences schemes.

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Notes

  1. Although unlikely to occur in practice, we do not exclude this possibility.

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  1. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC - UFABC, Rua Santa Adélia, 166, bairro Bangu, Santo André, SP, 09210-170, Brazil

    André Pierro de Camargo

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  1. André Pierro de Camargo
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Correspondence to André Pierro de Camargo.

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de Camargo, A. Rounding error analysis of divided differences schemes: Newton’s divided differences; Neville’s algorithm; Richardson extrapolation; Romberg quadrature; etc.. Numer Algor 85, 591–606 (2020). https://doi.org/10.1007/s11075-019-00828-1

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  • DOI: https://doi.org/10.1007/s11075-019-00828-1

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