Complex Gaussian quadrature of oscillatory integrals |
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Authors: | Alfredo Deaño Daan Huybrechs |
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Affiliation: | (1) DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK;(2) Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium |
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Abstract: | We construct and analyze Gauss-type quadrature rules with complex- valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behaviour, suitable for the application of Gaussian rules with non-standard weight functions. The results differ from those in previous research in the sense that the constructed rules are asymptotically optimal, i.e., among all known methods for oscillatory integrals they deliver the highest possible asymptotic order of convergence, relative to the required number of evaluations of the integrand. |
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Keywords: | KeywordHeading" >Mathematics Subject Classification (2000) Primary 65D30 Secondary 30E20 41A60 |
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