Computation of irregularly oscillating integrals

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 int egrands with a constant period and integrands with an increasingly rapid os cillatory behaviour at infinity. If the phase function of the oscillating f actor 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 no nlinear phase function, the point where the extrapolation process is to be started must be selected carefully, since maxima in the phase function indu ce steps in the sequence of partial sums. So far, this problem has been neg lected in the literature. Almost no off-the-shelf implementations of algorithms suitable for the comp utation 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 Delta (2)-process, and the epsilon -algorithm. Th ey are compared with Sidi's W-transformation that has been devised to evalu ate such integrals. The algorithms have been tested with several benchmarks including problems with linear and polynomial phase functions as well as c ases appropriate for asymptotic partitioning. Although designed for the lat ter, 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 epsilon -algorithm yields the best overall results. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.