Asymptotic properties of minimal integration rules

Rabinowitz, Philip; Richter-Dyn, Nira

doi:10.1090/S0025-5718-1970-0298946-X

Asymptotic properties of minimal integration rules

Authors: Philip Rabinowitz and Nira Richter-Dyn
Journal: Math. Comp. 24 (1970), 593-609
MSC: Primary 65D30
DOI: https://doi.org/10.1090/S0025-5718-1970-0298946-X
MathSciNet review: 0298946
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The error of a particular integration rule applied to a Hilbert space of functions analytic within an ellipse containing the interval of integration is a bounded linear functional. Its norm, which depends on the size of the ellipse, has proved useful in estimating the truncation error occurring when the integral of a particular analytic function is approximated using the rule in question. It is thus of interest to study rules which minimize this norm, namely minimal integration rules. The present paper deals with asymptotic properties of such minimal integration rules as the underlying ellipses shrink to the interval of integration.

[1] R. E. Barnhell, "Asymptotic properties of minimal norm and optimal quadrature," Numer. Math., v. 12, 1968, pp. 384-393.
[2] Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR 0157156
[3] R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization, Comput. J. 6 (1963/64), 163–168. MR 152116, https://doi.org/10.1093/comjnl/6.2.163
[4] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Fourth edition prepared by Ju. V. Geronimus and M. Ju. Ceĭtlin. Translated from the Russian by Scripta Technica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York-London, 1965. MR 0197789
[5] Philip Rabinowitz and Nira Richter, New error coefficients for estimating quadrature errors for analytic functions, Math. Comp. 24 (1970), 561–570. MR 275675, https://doi.org/10.1090/S0025-5718-1970-0275675-X
[6] Nira Richter, Properties of minimal integration rules, SIAM J. Numer. Anal. 7 (1970), 67–79. MR 260176, https://doi.org/10.1137/0707003
[7] E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
[8] R. A. Valentin, Applications of Functional Analysis to Optimal Numerical Approximations for Analytic Functions, Ph.D. Thesis, Brown University, Providence, R.I., 1965.