×

Asymptotisches Verhalten und Konvergenzbeschleunigung von Iterationsfolgen. (German) Zbl 0608.65004

General procedures for accelerating convergence of fixed point methods by extrapolation are discussed. An asymptotic expansion for such iterative sequences is proved. It is shown that the behaviour of this expansion, neglecting the remainder, is closely related to the asymptotic behaviour of sequences satisfying a linear difference equation. A discussion of Wynn’s \(\epsilon\)-algorithm is given using the theory developed.
Reviewer: T.Håvie

MSC:

65B05 Extrapolation to the limit, deferred corrections
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Brezinski, C.: Accélération de la convergence en analyse numérique. Lect. Notes Math.584, 96-125 (1977)
[2] Brezinski, C.: A general extrapolation algorithm. Numer. Math.35, 175-187 (1980) · Zbl 0444.65001
[3] de Bruijn, N.G.: Asymptotic methods in analysis. Amsterdam: North-Holland 1958 · Zbl 0082.04202
[4] Dieudonné, J.: Grundzüge der modernen Analysis. Berlin: VEB Deutscher Verlag der Wissenschaften 1971 · Zbl 0208.31801
[5] Faddejew, D.K., Faddejewa, W.N.: Numerische Methoden der linearen Algebra. Berlin: VEB Deutscher Verlag der Wissenschaften 1964 · Zbl 0119.12202
[6] Griepentrog, E.: Eine Modifikation des Minimalpolynomverfahrens zur Lösung von Fixpunktgleichungen. Wiss. Informationen TH Karl-Marx-Stadt29, 7 (1981)
[7] Meinardus, G.: Über das asymptotische Verhalten von Iterationsfolgen. Z. Angew. Math. Mech.63, 70-72 (1983) · Zbl 0519.65004
[8] Meschkowski, H.: Differenzengleichungen. Göttingen: Vandenhoek & Ruprecht 1959
[9] Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. New York, London: Academic Press 1970 · Zbl 0241.65046
[10] Ortloff, H.: Konvergenzbeschleunigung iterativer Algorithmen durch Extrapolationsmethoden. Diss. A TH Magdeburg 1983
[11] Schmidt, J.W.: Asymptotische Einschließungen bei konvergenzbeschleunigenden Verfahren. Numer. Math.8, 105-113 (1966) · Zbl 0135.37903
[12] Shanks, D.: Nonlinear transformations of divergent and slowly convergent sequences. J. Math. Phys.34, 1-42 (1955) · Zbl 0067.28602
[13] Skelboe, S.: Computation of the periodic steady-state response of nonlinear networks by extrapolation methods. IEEE Trans. CAS27, 161-175 (1980) · Zbl 0431.94045
[14] Wynn, P.: On a device for computing thee m(Sn-transformation. MTAC10, 91-96 (1956) · Zbl 0074.04601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.