Numerical Integration of Functions of a Rapidly Rotating Phase

Kurzfassung/Abstract

We present an algorithm for the efficient numerical evaluation of integrals of the form $I(\omega) = \int_0¹ F( x,\e^{\i \omega x}; \omega) \, \d x$ for sufficiently smooth but otherwise arbitrary $F$ and $\omega \gg 1$. The method is entirely “black-box,” i.e., it does not require the explicit computation of moment integrals or other precomputations 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.