Summary
For the problem of continuing a given Taylor series beyond the circle of convergence a classical summation method is proposed; numerical properties are studied. A known theorem about the “region of continuation” yields results about the speed of convergence of the transformed series. For special problems there are described in some sense optimal methods. Recursion formulas for the computation of the summation matrix are given.
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Literatur
Henrici, P.: An algorithm for analytic continuation. J. SIAM Numer. Anal.3, 67–78 (1966).
Hille, E.: Analytic function theory II. Boston-New York: Ginn & Co. 1962.
Knopp, K.: Über Polynomentwicklungen im Mittag-Lefflerschen Stern durch Anwendung der Eulerschen Reihentransformation. Acta Math.47, 313–335 (1926).
Kublanowskaya, V. N.: Application of analytic continuation in numerical analysis by means of change of variables. Trudy Mat. Inst. Steklov53, 145–185 (1959).
Niethammer, W.: Konvergenzbeschleunigung bei einstufigen Iterationsverfahren durch Summierungsmethoden. Iterationsverfahren, Numerische Mathematik, Approximationstheorie. ISNM Vol. 15, S. 235–243. Basel-Stuttgart: Birkhäuser 1970.
Niethammer, W., Schempp, W.: On the construction of iteration methods for linear equations in Banach spaces by summation methods. Aequat. Math.5, 273–284 (1970).
Perron, O.: Über eine Verallgemeinerung der Eulerschen Reihentransformation. Math. Zeitschr.18, 157–172 (1923).
Zeller, K., Beekmann, W.: Theorie der Limitierungsverfahren. Berlin-Heidelberg-New York: Springer 1970.
Zelmer, G.: Summation methods in the two-and three-body problems. Thesis. University of British-Columbia, May 1967.
Zelmer, G.: (E, ψ, α, β) Summability and applications. Arch. Rat. Mech. Anal.35, 211–219 (1969).
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Niethammer, W. Ein numerisches Verfahren zur analytischen Fortsetzung. Numer. Math. 21, 81–92 (1973). https://doi.org/10.1007/BF01436189
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DOI: https://doi.org/10.1007/BF01436189