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H. Mohamad and M. Oliver,
Numerical integration of functions of a rapidly rotating
phase,
SIAM J. Num. Anal. 59 (2021), 2310-2319.
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.
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