Numerical integration of functions of a rapidly rotating phase

Abstract

We present an algorithm for the efficient numerical evaluation of integrals of the form \[ I(\omega) = \int_0^1 F( x,\mathrm e^{\mathrm i \omega x}; \omega) \, \mathrm d x \] for sufficiently smooth but otherwise arbitrary $F$ and $\omega \gg 1$. 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 $\omega$. 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.