Geodesic
Addition to the Aitken method for the extrapolation of the limit of slowly convergent sequences
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 2, pp. 193-201 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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V. N. Bakulin; V. V. Inflianskas. Addition to the Aitken method for the extrapolation of the limit of slowly convergent sequences. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 2, pp. 193-201. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_2_a1/

[1] Knopp K., Theory and application of infinite series, 2nd English ed., Dover Publ., New York, 1990

[2] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, v. 2, Nauka, M., 1969

[3] NIST Handbook of mathematical functions, Cambridge University Press, 2010 | MR | Zbl

[4] Brezinski C., Redivo Zaglia M., Extrapolation Methods. Theory and Practice, North-Holland Publishing Co, 1991 | MR | Zbl

[5] Sidi A., Practical Extrapolation Methods, Cambridge University Press, 2003 | MR | Zbl

[6] Aitken A. C., “On Bernoulli's numerical solution of algebraic equations”, Proc. Roy. Soc. Edinburgh. Ser. A, 46 (1926), 289–305 | DOI | Zbl

[7] Krylov V. I., Bobkov V. V., Monastyrskii P. I., Vychislitelnye metody, v. 2, Nauka, M., 1977

[8] Lubkin S., “A method of summing infinite series”, J. Res. Nat. Bur. Stand., 48 (1952), 228–254 | DOI | MR

[9] Bromwich T. J., An introduction to the theory of infinite series, 3rd ed., Chelsea, New York, 1991 | Zbl

[10] Henrici P., Elements of numerical analysis, Wiley, New-York, l964 | MR | Zbl

[11] Wimp J., Sequence transformations and their applications, Acad. Press, New York, 1981 | MR | Zbl

[12] Clark W. D., Gray H. L., Adams J. E., “A note on the T-transformation of Lubkin”, J. Res. Nat. Bur. Stand., 73B:1 (1968), 25–29 | DOI | MR

[13] Tucker R. R., “The $\delta^2$-process and related topics”, Pacific J. Math., 22:2 (1967), 349–359 | DOI | MR | Zbl

[14] Tucker R. R., “The $\delta^2$-process and related topics, II”, Pacific J. Math., 28a:2 (1969), 455–463 | DOI | MR | Zbl

[15] Overholt K. J., “Extended Aitken acceleration”, BIT, 5 (1965), 122–132 | DOI | MR | Zbl

[16] Drummond J. E., “Summing a common type of slowly convergent series of positive terms”, J. Austral. Math. Soc. Ser. B, 19 (1976), 416–421 | DOI | MR | Zbl

[17] Bjorstad P., Dahlquist G., Grosse E., “Extrapolation of asymptotic expansions by a modified Aitken $\delta^2$-formula”, BIT, 21 (1981), 56–65 | DOI | MR | Zbl

[18] Schmidt R. J., “On the numerical solution of linear simultaneous equations by an iterative method”, Phil. Mag., 32 (1941), 369–383 | DOI | MR | Zbl

[19] Shanks D., “Non-linear transformations of divergent and slowly convergent sequences”, J. Math. Phys., 34 (1955), 1–42 | DOI | MR | Zbl

[20] Wynn P., “On a device for computing the $\mathrm{e_m(S_n)}$ transformation”, MTAC, 10 (1956), 91–96 | MR | Zbl

[21] Gray H. L., Atchison T. A., “Nonlinear transformations related to the evaluation of improper integrals, I”, SIAM J. Numer. Analys., 1967, no. 4, 363–371 | DOI | MR | Zbl

[22] Levin D., “Development of non-linear transformations for improving convergence of sequences”, Internat. J. Comput. Math., 1973, no. 3, 371–388 | MR | Zbl

[23] Havie T., “Generalized Neville type extrapolation schemes”, BIT, 19 (1979), 204–213 | DOI | MR | Zbl

[24] Brezinski C., “A general extrapolation algorithm”, Numer. Math., 35 (1980), 175–187 | DOI | MR | Zbl

[25] Richardson L. F., “The deferred approach to the limit. Part I: Single lattice”, Philos. Trans. Roy. Soc. London. Ser A, 226 (1927), 299–349 | DOI | Zbl

[26] Marchuk G. I., Shaidurov V. V., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979 | MR

[27] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, v. 1, Nauka, M., 1969