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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.


References [Enhancements On Off] (What's this?)

  • [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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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1988-0942153-5
Article copyright: © Copyright 1988 American Mathematical Society

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