Abstract
The class of sequences and series in which the Aitken process accelerates the convergence is considerably extended. It is proved that a proper subsequence of a slowly convergent sequence satisfies the sufficient condition for accelerating the convergence of the Aitken transformation. Two numerical examples illustrate the highly accurate limit extrapolation.
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References
K. Knopp, Theory and Application of Infinite Series, 2nd ed. (Dover, New York, 1990).
G. M. Fikhtengolts, The Fundamentals of Mathematical Analysis, Vol. 2 (Nauka, Moscow, 1969; Pergamon, Oxford, 1965).
NIST Handbook of Mathematical Functions (Cambridge Univ. Press, 2010).
C. Brezinski and M. Redivo Zaglia, Extrapolation Methods: Theory and Practice (North-Holland, 1991).
A. Sidi, Practical Extrapolation Methods (Cambridge Univ. Press, 2003).
A. C. Aitken, “On Bernoulli’s numerical solution of algebraic equations,” Proc. R. Soc. Edinburgh Ser. A, No. 46, 289–305 (1926).
V. I. Krylov, V. V. Bobkov, and P. I. Monastyrnyi, Numerical Methods, Vol. 2 (Nauka, Moscow, 1977) [in Russian].
S. Lubkin, “A method of summing infinite series,” J. Res. Nat. Bur. Stand., No. 48, 228–254 (1952).
T. J. Bromwich, An Introduction to the Theory of Infinite Series, 3rd ed. (Chelsea, New York, 1991).
P. Henrici, Elements of Numerical Analysis (Wiley, New York, 1964).
J. Wimp, Sequence Transformations and Their Applications (Academic, New York, 1981).
W. D. Clark, H. L. Gray, and J. E. Adams, “A note on the T-transformation of Lubkin,” J. Res. Nat. Bur. Stand. 73B (1), 25–29 (1968).
R. R. Tucker, “The d2-process and related topics,” Pacific J. Math. 22, 349–359 (1967).
R. R. Tucker, “The d2-process and related topics II,” Pacific J. Math. 28, 455–463 (1969).
K. J. Overholt, “Extended Aitken acceleration,” BIT, No. 5, 122–132 (1965).
J. E. Drummond, “Summing a Common Type of Slowly Convergent Series of Positive Terms,” J. Austral. Math. Soc. Ser. B, No. 19, 416–421 (1976).
P. Bjørstad, G. Dahlquist, and E. Grosse, “Extrapolation of asymptotic expansions by a modified Aitken d2-formula,” BIT, No. 21, 56–65 (1981).
R. J. Schmidt, “On the numerical solution of linear simultaneous equations by an iterative method,” Philos. Mag., No. 32, 369–383 (1941).
D. Shanks, “Non-Linear Transformations of Divergent and Slowly Convergent Sequences,” J. Math. Phys., No. 34, 1–42 (1955).
P. Wynn, “On a device for computing the em(Sn) transformation,” MTAC, No. 10, 91–96 (1956).
H. L. Gray and T. A. Atchison, “Nonlinear transformations related to the evaluation of improper integrals I,” SIAM J. Numer. Analys., No. 4, 363–371 (1967).
D. Levin, “Development of non-linear transformations for improving convergence of sequences,” Int. J. Comput. Math., No. 3, 371–388 (1973).
T. Håvie, “Generalized Neville type extrapolation schemes,” BIT, No. 19, 204–213 (1979).
C. Brezinski, “A general extrapolation algorithm,” Numer. Math., ? 35, 175–187 (1980).
L. F. Richardson, “The deferred approach to the limit. Part I Single lattice,” Philos. Trans. R. Soc. London, Ser A, 226, 299–349 (1927).
G. I. Marchuk and V. V. Shaidurov, Improving the Accuracy of Difference Scheme Solutions (Nauka, Moscow, 1979) [in Russian].
G. M. Fikhtengolts, The Fundamentals of Mathematical Analysis, Vol. 1 (Nauka, Moscow, 1969; Pergamon, Oxford, 1965).
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Original Russian Text © V.N. Bakulin, V. Inflianskas, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 2, pp. 193–201.
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Bakulin, V.N., Inflianskas, V. Addition to the Aitken method for the extrapolation of the limit of slowly convergent sequences. Comput. Math. and Math. Phys. 56, 191–199 (2016). https://doi.org/10.1134/S0965542516020044
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DOI: https://doi.org/10.1134/S0965542516020044