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IMA Journal of Numerical Analysis 2005 25(1):25-44; doi:10.1093/imanum/drh022
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IMA Journal of Numerical Analysis Vol. 25 No. 1 © Institute of Mathematics and its Applications 2005; all rights reserved

On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators

Arieh Iserles1 *

1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Rd, Cambridge CB3 0WA, UK

In this paper we set out to understand Filon-type quadrature of highly-oscillating integrals of the form {int}01 f(x) ei{omega}g(x) dx, where g is a real-valued function and {omega} >> 1. Employing ad hoc analysis, as well as perturbation theory, we demonstrate that for most functions g of interest the moments behave asymptotically according to a specific model that allows for an optimal choice of quadrature nodes. Filon-type methods that employ such quadrature nodes exhibit significantly faster decay of the error for high frequencies {omega}. Perhaps counterintuitively, as long as optimal quadrature nodes are used, rapid oscillation leads to significantly more precise and more affordable quadrature.

Key Words: numerical quadrature; high oscillation; asymptotic expansions; irregular oscillators

Received 16 October 2003. Revised 5 August 2004.

* Email: ai{at}damtp.cam.ac.uk


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A. Iserles and S. P Norsett
Efficient quadrature of highly oscillatory integrals using derivatives
Proc R Soc A, May 8, 2005; 461(2057): 1383 - 1399.
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