A user-friendly extrapolation method for oscillatory infinite integrals

Author:
Avram Sidi

Journal:
Math. Comp. **51** (1988), 249-266

MSC:
Primary 65D30; Secondary 41A55, 65B05

DOI:
https://doi.org/10.1090/S0025-5718-1988-0942153-5

MathSciNet review:
942153

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Abstract: In a recent publication [4] the author developed an extrapolation method, the *W*-transformation, for the accurate computation of convergent oscillatory infinite integrals. In yet another publication [6] this method was shown to be applicable to divergent oscillatory infinite integrals that are defined in the sense of summability. The application of the *W*-transformation involves some asymptotic analysis of the integrand as the variable of integration tends to infinity. In the present work the *W*-transformation is modified so as to keep this asymptotic analysis to a minimum, involving only the phase of oscillations. This modified version, which turns out to be as efficient as the original *W*-transformation, can also be applied to convergent or divergent oscillatory infinite integrals other than those dealt with in [4] and [6]. The convergence properties of the modified transformation are analyzed in detail for the integrals of [4] and [6], and numerical examples are provided.

**[1]**I. M. Longman, "Note on a method for computing infinite integrals of oscillatory functions,"*Proc. Cambridge Philos. Soc.*, v. 52, 1956, pp. 764-768. MR**0082193 (18:515f)****[2]**R. E. Powell & S. M. Shah,*Summability Theory and its Applications*, Van Nostrand Reinhold, London, 1972.**[3]**A. Sidi, "Extrapolation methods for oscillatory infinite integrals,"*J. Inst. Math. Appl.*, v. 26, 1980, pp. 1-20. MR**594340 (81m:40002)****[4]**A. Sidi, "The numerical evaluation of very oscillatory infinite integrals by extrapolation,"*Math. Comp.*, v. 38, 1982, pp. 517-529. MR**645667 (83i:65023)****[5]**A. Sidi, "An algorithm for a special case of a generalization of the Richardson extrapolation process,"*Numer. Math.*, v. 38, 1982, pp. 299-307. MR**654099 (83d:65012)****[6]**A. Sidi, "Extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability,"*J. Comput. Appl. Math.*, v. 17, 1987, pp. 105-114. MR**884264 (88a:65025)**

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DOI:
https://doi.org/10.1090/S0025-5718-1988-0942153-5

Article copyright:
© Copyright 1988
American Mathematical Society