On the numerical quadrature of highly-oscillating integrals II: Irregular oscillators |
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Authors: | Iserles Arieh |
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Institution: |
1 Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Rd, Cambridge CB3 0WA, UK
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Abstract: | In this paper we set out to understand Filon-type quadratureof highly-oscillating integrals of the form 01 f(x) eig(x) dx,where g is a real-valued function and >> 1. Employingad hoc analysis, as well as perturbation theory, we demonstratethat for most functions g of interest the moments behave asymptoticallyaccording to a specific model that allows for an optimal choiceof quadrature nodes. Filon-type methods that employ such quadraturenodes exhibit significantly faster decay of the error for highfrequencies . Perhaps counterintuitively, as long as optimalquadrature nodes are used, rapid oscillation leads to significantlymore precise and more affordable quadrature. |
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Keywords: | numerical quadrature high oscillation asymptotic expansions irregular oscillators |
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