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2013.28: Efficient high-order rational integration and deferred correction with equispaced data

2013.28: Stefan Güttel and Georges Klein (2013) Efficient high-order rational integration and deferred correction with equispaced data.

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Abstract

Stable high-order linear interpolation schemes are well suited for the accurate approximation of antiderivatives and the construction of efficient quadrature rules. In this paper we utilize for this purpose the family of linear barycentric rational interpolants by Floater and Hormann, which are particularly useful for interpolation with equispaced nodes. We analyze the convergence of integrals of these interpolants to those of analytic functions as well as functions with a finite number of continuous derivatives. As a by-product, our convergence analysis leads to an extrapolation scheme for rational quadrature at equispaced nodes. Furthermore, as a main application of our analysis, we present and investigate a new iterated deferred correction method for the solution of initial value problems, which allows to work efficiently even with large numbers of equispaced data. This so-called rational deferred correction (RDC) method turns out to be highly competitive with other methods relying on more involved implementations or non-equispaced node distributions. Extensive numerical experiments are carried out, comparing the RDC method to the well established spectral deferred correction method by Dutt, Greengard and Rokhlin.

Item Type:MIMS Preprint
Uncontrolled Keywords:quadrature, barycentric rational interpolation, extrapolation, deferred correction
Subjects:MSC 2000 > 41 Approximations and expansions
MSC 2000 > 65 Numerical analysis
MIMS number:2013.28
Deposited By:Stefan Güttel
Deposited On:24 May 2013

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