The numerical evaluation of very oscillatory infinite integrals by extrapolation
Author:
Avram Sidi
Journal:
Math. Comp. 38 (1982), 517529
MSC:
Primary 65D30; Secondary 41A55, 65B99
MathSciNet review:
645667
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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, McGrawHill
Book Company, Inc., New YorkTorontoLondon, 1956. MR 0075670
(17,788d)
 [2]
David
Levin, Development of nonlinear transformations of improving
convergence of sequences, Internat. J. Comput. Math.
3 (1973), 371–388. MR 0359261
(50 #11716)
 [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 0483313
(58 #3327)
 [4]
D. Levin & A. Sidi, "Two new classes of nonlinear 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. 224231.
 [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. 429440.
 [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 0203312
(34 #3165)
 [9]
Avram
Sidi, Convergence properties of some
nonlinear sequence transformations, Math.
Comp. 33 (1979), no. 145, 315–326. MR 514827
(81h:65003), http://dx.doi.org/10.1090/S00255718197905148276
 [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
(81a:65011)
 [11]
Avram
Sidi, Analysis of convergence of the
𝑇transformation for power series, Math. Comp. 35 (1980), no. 151, 833–850. MR 572860
(83d:41039), http://dx.doi.org/10.1090/S00255718198005728600
 [12]
A.
Sidi, Extrapolation methods for oscillatory infinite
integrals, J. Inst. Math. Appl. 26 (1980),
no. 1, 1–20. MR 594340
(81m:40002)
 [13]
A. Sidi, "An algorithm for a special case of a generalization of the Richardson extrapolation process," Numer. Math. (In press.)
 [1]
 F. B. Hildebrand, Introduction to Numerical Analysis, McGrawHill, New York, 1956. MR 0075670 (17:788d)
 [2]
 D. Levin, "Development of nonlinear transformations for improving convergence of sequences," Internat. J. Comput. Math., v. B3, 1973, pp. 371388. MR 0359261 (50:11716)
 [3]
 D. Levin, "Numerical inversion of the Laplace transform by accelerating the convergence of Bromwich's integral," J. Comput. Appl. Math., v. 1, 1975, pp. 247250. MR 0483313 (58:3327)
 [4]
 D. Levin & A. Sidi, "Two new classes of nonlinear 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. 224231.
 [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. 429440.
 [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., v. 3, 1966, pp. 390409. MR 0203312 (34:3165)
 [9]
 A. Sidi, "Convergence properties of some nonlinear sequence transformations," Math. Comp., v. 33, 1979, pp. 315326. MR 514827 (81h:65003)
 [10]
 A. Sidi, "Some properties of a generalization of the Richardson extrapolation process," J. Inst. Math. Appl., v. 24, 1979, pp. 327346. MR 550478 (81a:65011)
 [11]
 A. Sidi, "Analysis of convergence of the Ttransformation for power series," Math. Comp., v. 35, 1980, pp. 833850. MR 572860 (83d:41039)
 [12]
 A. Sidi, "Extrapolation methods for oscillatory infinite integrals," J. Inst. Math. Appl., v. 26, 1980, pp. 120. MR 594340 (81m:40002)
 [13]
 A. Sidi, "An algorithm for a special case of a generalization of the Richardson extrapolation process," Numer. Math. (In press.)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65D30,
41A55,
65B99
Retrieve articles in all journals
with MSC:
65D30,
41A55,
65B99
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206456675
PII:
S 00255718(1982)06456675
Article copyright:
© Copyright 1982
American Mathematical Society
