A userfriendly extrapolation method for oscillatory infinite integrals
Author:
Avram Sidi
Journal:
Math. Comp. 51 (1988), 249266
MSC:
Primary 65D30; Secondary 41A55, 65B05
MathSciNet review:
942153
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Abstract 
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Abstract: In a recent publication [4] the author developed an extrapolation method, the Wtransformation, 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 Wtransformation involves some asymptotic analysis of the integrand as the variable of integration tends to infinity. In the present work the Wtransformation 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 Wtransformation, 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
(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. 26 (1980),
no. 1, 1–20. MR 594340
(81m:40002)
 [4]
Avram
Sidi, The numerical evaluation of very
oscillatory infinite integrals by extrapolation, Math. Comp. 38 (1982), no. 158, 517–529. MR 645667
(83i:65023), http://dx.doi.org/10.1090/S00255718198206456675
 [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
(83d:65012), http://dx.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. 12, 105–114. MR 884264
(88a:65025), http://dx.doi.org/10.1016/03770427(87)900410
 [1]
 I. M. Longman, "Note on a method for computing infinite integrals of oscillatory functions," Proc. Cambridge Philos. Soc., v. 52, 1956, pp. 764768. 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. 120. MR 594340 (81m:40002)
 [4]
 A. Sidi, "The numerical evaluation of very oscillatory infinite integrals by extrapolation," Math. Comp., v. 38, 1982, pp. 517529. 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. 299307. 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. 105114. MR 884264 (88a:65025)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809421535
PII:
S 00255718(1988)09421535
Article copyright:
© Copyright 1988
American Mathematical Society
