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.**52**(1956), 764–768. MR**0082193****[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.**26**(1980), no. 1, 1–20. MR**594340****[4]**Avram Sidi,*The numerical evaluation of very oscillatory infinite integrals by extrapolation*, Math. Comp.**38**(1982), no. 158, 517–529. MR**645667**, https://doi.org/10.1090/S0025-5718-1982-0645667-5**[5]**Avram Sidi,*An algorithm for a special case of generalization of the Richardson extrapolation process*, Numer. Math.**38**(1981/82), no. 3, 299–307. MR**654099**, https://doi.org/10.1007/BF01396434**[6]**Avram Sidi,*Extrapolation methods for divergent oscillatory infinite integrals that are defined in the sense of summability*, J. Comput. Appl. Math.**17**(1987), no. 1-2, 105–114. MR**884264**, https://doi.org/10.1016/0377-0427(87)90041-0

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

Article copyright:
© Copyright 1988
American Mathematical Society