Computation of irregularly oscillating integrals Original Research Article

A suitable method to compute infinite integrals with oscillatory integrands is to partition the integration interval, for example at the zeros of the integrand. The limit of the resulting sequence of partial sums can then be found by means of extrapolation. This strategy is equally applicable to integrands with a constant period and integrands with an increasingly rapid oscillatory behaviour at infinity. If the phase function of the oscillating factor has a complicated form, its polynomial part can serve as a basis for an asymptotic partition which may be easier to compute. In the case of a nonlinear phase function, the point where the extrapolation process is to be started must be selected carefully, since maxima in the phase function induce steps in the sequence of partial sums. So far, this problem has been neglected in the literature. Almost no off-the-shelf implementations of algorithms suitable for the computation of irregularly oscillating integrals are available. Therefore, this article explores the usefulness of three standard acceleration algorithms that can be applied to a previously computed sequence of partial sums: the Euler transformation, the Δ2-process, and the ε-algorithm. They are compared with Sidiʹs W-transformation that has been devised to evaluate such integrals. The algorithms have been tested with several benchmarks including problems with linear and polynomial phase functions as well as cases appropriate for asymptotic partitioning. Although designed for the latter, the W-transformation is found to have a surprisingly poor performance if the difference between the phase function and its polynomial part is too large, whereas the ε-algorithm yields the best overall results.