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Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur

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Summary

Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

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Literaturverzeichnis

  1. H. Behnke undF. Sommer,Theorie der analytischen Funktionen einer komplexen Veränderlichen, 2. Aufl. (Springer-Verlag 1962).

  2. Ph. Davis,On the Numerical Integration of Periodic Analytic Functions. On Numerical Approximation, R. E. Langer ed. (The University of Wisconsin Press, Madison 1959), p. 45–59.

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  3. J. N. Lyness andC. B. Moler,Numerical Differentiation of Analytic Functions, SIAM J. Numer. Anal.4, 202–210 (1967).

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  4. J. N. Lyness,Differentiation Formulas for Analytic Functions, Math. Comp.22, 352–362 (1968).

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Authors and Affiliations

  1. Mathematisches Institut der Universität München, Deutschland

    Günther Hämmerlin

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  1. Günther Hämmerlin
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Meinem verehrten LehrerH. Görtler zur Vollendung seines 60. Lebensjahres gewidmet

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Hämmerlin, G. Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur. Journal of Applied Mathematics and Physics (ZAMP) 20, 681–687 (1969). https://doi.org/10.1007/BF01590624

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  • DOI: https://doi.org/10.1007/BF01590624

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