Abstract
In this paper we discuss a modification of Romberg's algorithm [1] for the numerical approximation of the integral
The idea behind the modification is to split the sequence of approximations calculated by the algorithm into two sequences, one which approximates the integral from above, and one which approximates it from below. Hence, at any step during the calculations we obtain both lower and upper bounds for the “true” value of the integral (1).
With the permission of the author Kubik [2] has published an ALGOL procedure using the method described in this paper.
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References
W. Romberg,Vereinfacte Numerische Integration, Det Kong. Norske Videnskabers Selskabs Forhandlinger, Bd. 28, Nr. 7, Trondheim 1955.
R. N. Kubik, Comm. ACM, Vol. 8, p. 381.
D. R. Hartree, Numerical Analysis, p. 115, Oxford at the Clarendon press, 1958.
E. T. Goodwin, The evaluation of integrals of the form\(\int_{ - \infty }^\infty {f(x)e^{ - x^2 } } dx\), Proc. Cambridge Phil. Soc. 45 (1949), p. 241–245.
A. M. Krasun and W. Prager,Remark on Romberg quadrature, Comm. ACM. 8, 4, 1965, p. 236.
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Håvie, T. On a modification of Romberg's algorithm. BIT 6, 24–30 (1966). https://doi.org/10.1007/BF01939546
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DOI: https://doi.org/10.1007/BF01939546