Asymptotic properties of minimal integration rules
Authors: Philip Rabinowitz and Nira Richter-Dyn
Journal: Math. Comp. 24 (1970), 593-609
MSC: Primary 65D30
MathSciNet review: 0298946
Full-text PDF Free Access
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.
-  R. E. Barnhell, "Asymptotic properties of minimal norm and optimal quadrature," Numer. Math., v. 12, 1968, pp. 384-393.
-  Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR 0157156
-  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
-  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
-  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
-  Nira Richter, Properties of minimal integration rules, SIAM J. Numer. Anal. 7 (1970), 67–79. MR 260176, https://doi.org/10.1137/0707003
-  E. C. Titchmarsh, Han-shu lun, Translated from the English by Wu Chin, Science Press, Peking, 1964 (Chinese). MR 0197687
-  R. A. Valentin, Applications of Functional Analysis to Optimal Numerical Approximations for Analytic Functions, Ph.D. Thesis, Brown University, Providence, R.I., 1965.
- R. E. Barnhell, "Asymptotic properties of minimal norm and optimal quadrature," Numer. Math., v. 12, 1968, pp. 384-393.
- P. J. Davis, Interpolation and Approximation, Blaisdell, Waltham, Mass., 1963. MR 28 #393. MR 0157156 (28:393)
- R. Fletcher & M. J. D. Powell, "A rapidly convergent descent method for minimization," Comput. J., v. 6, 1963/64, pp. 163-168. MR 27 #2096. MR 0152116 (27:2096)
- I. S. Gradshteyn & I. M. Ryzhtk, Tables of Integrals, Series, and Products, Fizmatgiz, Moscow, 1963; English transl., Academic Press, New York, 1965. MR 28 #5198; MR 33 #5952. MR 0197789 (33:5952)
- P. Rabinowitz & N. Richter, "New error coefficients for estimating quadrature error for analytic functions," Math. Comp., v. 24, 1970, pp. 561-570. MR 0275675 (43:1428)
- N. Richter, "Properties of minimal integration rules. II." (To appear.) MR 0260176 (41:4804)
- E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford Univ. Press, Oxford, 1939. MR 0197687 (33:5850)
- R. A. Valentin, Applications of Functional Analysis to Optimal Numerical Approximations for Analytic Functions, Ph.D. Thesis, Brown University, Providence, R.I., 1965.
Retrieve articles in Mathematics of Computation with MSC: 65D30
Retrieve articles in all journals with MSC: 65D30
Keywords: Hilbert space of analytic functions, norm of error functional, minimizing abscissae and weights, minimal norm integration rule, complete orthonormal set, asymptotic properties of minimal rules
Article copyright: © Copyright 1970 American Mathematical Society