PIER M
 
Progress In Electromagnetics Research M
ISSN: 1937-8726
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 14 > pp. 233-245

ACCELERATION OF SLOWLY CONVERGENT SERIES VIA THE GENERALIZED WEIGHTED-AVERAGES METHOD

By A. G. Polimeridis, R. M. Golubovic Niciforovi, and J. R. Mosig

Full Article PDF (169 KB)

Abstract:
A generalized version of the weighted-averages method is presented for the acceleration of convergence of sequences and series over a wide range of test problems, including linearly and logarithmically convergent series as well as monotone and alternating series. This method was originally developed in a partitionextrapolation procedure for accelerating the convergence of semiinfinite range integrals with Bessel function kernels (Sommerfeld-type integrals), which arise in computational electromagnetics problems involving scattering/radiation in planar stratified media. In this paper, the generalized weighted-averages method is obtained by incorporating the optimal remainder estimates already available in the literature. Numerical results certify its comparable and in many cases superior performance against not only the traditional weighted-averages method but also against the most proven extrapolation methods often used to speed up the computation of slowly convergent series.

Citation:
A. G. Polimeridis, R. M. Golubovic Niciforovi, and J. R. Mosig, "Acceleration of Slowly Convergent Series via the Generalized Weighted-Averages Method," Progress In Electromagnetics Research M, Vol. 14, 233-245, 2010.
doi:10.2528/PIERM10100702

References:
1. Weniger, E. J., "Nonlinear sequence transformations: Computational tools for the acceleration of convergence and the summation of divergent series,", preprint arXiv:math/0107080v1, http://arXiv.org..
doi:10.2528/PIER07052502

2. Valagiannopoulos, C. A., "An overview of the Watson transformation presented through a simple example," Progress In Electromagnetics Research, Vol. 75, 137-152, 2007.
doi:10.1017/S030500410003187X

3. Longman, I. M., "Note on a method for computing infinite integrals of oscillatory functions," Proc. Cambridge Phil. Soc., Vol. 52, 764-768, 1956.

4. I'A, T. J., An Introduction to the Theory of Infinite Series,, Macmillan, New York, 1965.
doi:10.1017/S0305004100044765

5. Scraton, R. E., "A note on the summation of divergent power series," Proc. Cambridge Phil. Soc., Vol. 66, 109-114, 1969.
doi:10.1093/comjnl/14.4.437

6. Wynn, P., "A note on the generalized Euler transformation," Computer J., Vol. 14, 441, 1971.
doi:10.1137/0716017

7. Smith, D. A. and W. F. Ford, "Acceleration of linear and logarithmic convergence," SIAM J. Num. Anal., Vol. 16, 223-240, 1979.
doi:10.1016/0377-0427(84)90017-7

8. Drummond, J. E., "Convergence speeding, convergence and summability," J. Comput. Appl. Math., Vol. 11, 145-159, 1984.

9. Shanks, D., "Nonlinear transformation of divergent and slowly convergent sequences," J. Math. Phys., Vol. 34, 1-42, 1955.
doi:10.1080/00207167308803075

10. Levin, D., "Development of nonlinear transformations for improving convergence of sequences," Int. J. Comput. Math. Section B, Vol. 3, 371-388, 1973.
doi:10.2307/2002183

11. Wynn, P., "On a device for computing the em (Sn) transformation," Math. Tables Aids Comput., Vol. 10, 91-96, 1956.
doi:10.1090/S0025-5718-1982-0645665-1

12. Smith, D. A. and W. F. Ford, "Numerical comparisons of nonlinear convergence accelerators," Math. Comput., Vol. 38, 481-499, 1982.

13. Brezinski, C. and M. R. Zaglia, Extrapolation Methods, Amsterdam, North-Holland, 1991.

14. Mosig, J. R. and F. E. Gardiol, "A dynamical radiation model for microstrip structures," Adv. Electron. Electron Phys., Vol. 59, 139-237, Academic Eds, New York, 1982.

15. Mosig, J. R. and F. E. Gardiol, "Analytical and numerical techniques in the Green's function treatment of microstrip antennas and scatterers," Proc. Inst. Elect. Eng., Vol. 130, 175-182, 1983.

16. Mosig, J. R., "Integral equation techniques," Numerical Techniques for Microwave and Millimeter-wave Passive Structures, 133-213.
doi:10.1109/8.725271

17. Michalski, K. A., "Extrapolation methods for Sommerfeld integraltails," IEEE Trans. Antennas and Propagat., Vol. 46, 1405-1418, 1998.
doi:10.2528/PIER08102405

18. Li, H., H.-G. Wang, and H. Zhang, "An improvement of the GeEsselle's method for the evaluation of the Green's functions in the shielded multilayered structures," Progress In Electromagnetics Research, Vol. 88, 149-161, 2008.
doi:10.2528/PIER10062310

19. Firuzeh, Z. H., G. A. E. Vandenbosch, R. Moini, S. H. H. Sadeghi, and R. Faraji-Dana, "Efficient evaluation of Green's functions for lossy half-space problems," Progress In Electromagnetics Research, Vol. 109, 139-157, 2010.
doi:10.1016/S0377-0427(00)00359-9

20. Homeier, H. H. H., "Scalar Levin-type sequence transformations," J. Comput. Appl. Math., Vol. 122, 81-147, 2000.
doi:10.2529/PIERS050104011634

21. Liu, P. and Z.-F. Li, "Efficient computation of Z-parameter for the rectangular planar circuit analysis," PIERS Online, Vol. 1, No. 5, 611-614, 2005.
doi:10.1109/8.817649

22. Fikioris, G., "An application of convergence acceleration methods," IEEE Trans. Antennas and Propagat., Vol. 47, 1758-1418, 1999.
doi:10.1007/BF01400966

23. Schneider, C., "Vereinfachte rekursionen zur Richardson-extrapolation in spezialfÄallen," Num. Math., Vol. 24, 177-184, 1975.
doi:10.1007/BF01930850

24. Håvie, T., "Generalized Neville type extrapolation schemes," BIT, Vol. 19, 204-213, 1979.
doi:10.1007/BF01396314

25. Brezinski, C., "A general extrapolation algorithm," Num. Math., Vol. 35, 175-180, 1980.

26. Sidi, A., "A user-friendly extrapolation method for oscillatory infinite integrals," Math. Comput., Vol. 51, 249-266, 1988.
doi:10.1145/356044.356051

27. Fessler, T., W. F. Ford, and D. A. Smith, "HURRY: An acceleration algorithm for scalar sequences and series," ACM Trans. Math. Software, Vol. 9, 346-354, 1983.


© Copyright 2010 EMW Publishing. All Rights Reserved