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BIT Numerical Mathematics

, Volume 44, Issue 4, pp 755–772 | Cite as

On Quadrature Methods for Highly Oscillatory Integrals and Their Implementation

  • A. Iserles
  • S. P. NØrsett
Article

Abstract

The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference approximations, with spacing inversely proportional to frequency. This renders the computation of error bounds even cheaper and, more importantly, leads to a new family of quadrature methods for highly oscillatory integrals that can attain arbitrarily high asymptotic order without computation of derivatives.

Keywords

high oscillation quadrature asymptotic expansions Filon’s integration error control 

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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUnited Kingdom
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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