Computation of Oscillatory Infinite Integrals by Extrapolation Methods

Abstract

In a recent work [2] a nonlinear extrapolation method, the D- transformation, was proposed, and this method proved to be very useful in accelerating the convergence of infinite integrals , a≥0, of different kinds. The D-transformation was analyzed for its convergence properties in [3] within the framework of the generalized Richardson extrapolation process. Two modifications of the D-transformation for oscillatory infinite integrals were proposed in [5], which were denoted the D— and D—transformations. Another modification, the W-transformation, useful for “very oscillatory” infinite integrals, was given in [6], and this modification was extended in [8] to divergent oscillatory infinite integrals that are defined in the sense of (Abel) summability. The W-transformation was modified significantly and made very user-friendly in [9], where a detailed convergence analysis for it is also given. For additional convergence results see [4] and [11]. The advantage of these modifications over the D-transformation is that they can achieve a given level of accuracy with considerably less computing than the D-transformation.