Abstract
It is proved that the error terms for both the ordinary and the modified Romberg algorithms may be expressed in terms of Bernoulli polynomials and their related periodic functions. The properties of the error terms may thus be described using the known properties of the Bernoulli polynomials and functions.
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References
W. Romberg,Vereinfachte numerische Integration, Det K. Norske Selsk. Forhandlinger, Bd. 28, No. 7 (Trondheim, 1955).
F. L. Bauer, H. Rutishauser and E. Stiefel,New Aspects in Numerical Quadrature, Proc. Symp. Appl. Math. (AMS 1963), Vol. XV, pp. 198–218.
T. Håvie,On a Modification of Romberg's Algorithm, BIT (6) (1966), pp. 24–30.
T. Håvie,On the Practical Application of the Modified Romberg Algorithm, BIT (7) (1967), pp. 103–113.
T. Ström,Strict Error bounds in Romberg Quadrature, BIT (7) (1967), pp. 314–321.
R. Bulirsch and J. Stoer,Asymptotic upper and lower bounds for results of extrapolation methods, Numer. Math. (8) (1966), pp. 93–104.
J. W. Schmidt,Asymptotische Einschliessung bei konvergenzbeschleunigenden Verfahren. II, Numer. Math. (11) (1968), pp. 53–56.
Handbook of Mathematical Functions. (Edited by M. Abramowitz and A. Stegun). (Dover Publications, Inc., New York, 1965).
V. I. Krylov,Approximate Calculation of Integrals. (Macmillan, New York and London, 1962).
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On leave of absence from Kjeller Computer Installation, Norway
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Håvie, T. Derivation of explicit expressions for the error terms in the ordinary and the modified Romberg algorithms. BIT 9, 18–29 (1969). https://doi.org/10.1007/BF01933536
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DOI: https://doi.org/10.1007/BF01933536