Numerical integration of functions of a rapidly rotating phase

09/10/2019

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by   Haidar Mohamad, et al.

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We present an algorithm for the efficient numerical evaluation of integrals of the form I(ω) = ∫_0^1 F( x,e^iω x; ω) d x for sufficiently smooth but otherwise arbitrary F and ω≫ 1. The method is entirely "black-box", i.e., does not require the explicit computation of moment integrals or other pre-computations involving F. Its performance is uniform in the frequency ω. We prove that the method converges exponentially with respect to its order when F is analytic and give a numerical demonstration of its error characteristics.