The numerical evaluation of very oscillatory infinite integrals by extrapolation

Author:
Avram Sidi

Journal:
Math. Comp. **38** (1982), 517-529

MSC:
Primary 65D30; Secondary 41A55, 65B99

DOI:
https://doi.org/10.1090/S0025-5718-1982-0645667-5

MathSciNet review:
645667

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Abstract: Recently the author has given two modifications of a nonlinear extrapolation method due to Levin and Sidi, which enable one to accurately and economically compute certain infinite integrals whose integrands have a simple oscillatory behavior at infinity. In this work these modifications are extended to cover the case of very oscillatory infinite integrals whose integrands have a complicated and increasingly rapid oscillatory behavior at infinity. The new method is applied to a number of complicated integrals, among them the solution to a problem in viscoelasticity. Some convergence results for this method are presented.

**[1]**F. B. Hildebrand,*Introduction to numerical analysis*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. MR**0075670****[2]**David Levin,*Development of non-linear transformations of improving convergence of sequences*, Internat. J. Comput. Math.**3**(1973), 371–388. MR**359261**, https://doi.org/10.1080/00207167308803075**[3]**David Levin,*Numerical inversion of the Laplace transform by accelerating the convergence of Bromwich’s integral*, J. Comput. Appl. Math.**1**(1975), no. 4, 247–250. MR**483313**, https://doi.org/10.1016/0771-050X(75)90015-7**[4]**D. Levin & A. Sidi, "Two new classes of non-linear transformations for accelerating the convergence of infinite integrals and series,"*Appl. Math. Comput.*(In press.)**[5]**I. M. Longman, "Numerical Laplace transform inversion of a function arising in viscoelasticity,"*J. Comput. Phys.*, v. 10, 1972, pp. 224-231.**[6]**I. M. Longman, "On the generation of rational approximations for Laplace transform inversion with an application to viscoelasticity,"*SIAM J. Appl. Math.*, v. 24, 1973, pp. 429-440.**[7]**I. M. Longman, private communication, 1979.**[8]**G. F. Miller,*On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation*, SIAM J. Numer. Anal.**3**(1966), 390–409. MR**203312**, https://doi.org/10.1137/0703034**[9]**Avram Sidi,*Convergence properties of some nonlinear sequence transformations*, Math. Comp.**33**(1979), no. 145, 315–326. MR**514827**, https://doi.org/10.1090/S0025-5718-1979-0514827-6**[10]**A. Sidi,*Some properties of a generalization of the Richardson extrapolation process*, J. Inst. Math. Appl.**24**(1979), no. 3, 327–346. MR**550478****[11]**Avram Sidi,*Analysis of convergence of the 𝑇-transformation for power series*, Math. Comp.**35**(1980), no. 151, 833–850. MR**572860**, https://doi.org/10.1090/S0025-5718-1980-0572860-0**[12]**A. Sidi,*Extrapolation methods for oscillatory infinite integrals*, J. Inst. Math. Appl.**26**(1980), no. 1, 1–20. MR**594340****[13]**A. Sidi, "An algorithm for a special case of a generalization of the Richardson extrapolation process,"*Numer. Math.*(In press.)

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DOI:
https://doi.org/10.1090/S0025-5718-1982-0645667-5

Article copyright:
© Copyright 1982
American Mathematical Society