×

A nodal spline interpolant for the Gregory rule of even order. (English) Zbl 0793.41026

Summary: The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature, not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C. H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.

MSC:

41A55 Approximate quadratures
41A15 Spline approximation
41A05 Interpolation in approximation theory
65D32 Numerical quadrature and cubature formulas
65D30 Numerical integration
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Atkinson, K.E. (1978): An Introduction to Numerical Analysis., Wiley, New York · Zbl 0402.65001
[2] Babu?ka, I. (1966): Über die optimale Berechnung der Fourierschen Koeffizienten, Apl. Mat.11, 113-121 · Zbl 0171.37502
[3] Brass, H. (1977): Quadraturverfahren. Vandenhoeck & Ruprecht, Göttingen
[4] Dahmen, W., Goodman, T.N.T., Micchelli, C.A. (1988): Compactly supported fundamental functions for spline interpolation, Numer Math52, 639-664 · Zbl 0646.65012 · doi:10.1007/BF01395816
[5] Davis, P.J., Rabinowitz, P. (1984): Methods of Numerical Integration, Academic Press, Orlando · Zbl 0537.65020
[6] Delvos, F.-J. (1986): Interpolation of even periodic functions. In: C.K. Chui, L.L. Schumaker, J.D. Ward, eds., Approximation Theory V, pp. 315-318. Academic Press, Orlando · Zbl 0633.41003
[7] De Villiers, J.M. (1992): A convergence result in nodal spline interpolation, J. Approximation Theory (to appear) · Zbl 0790.41008
[8] De Villiers, J.M., Rohwer, C.H. (1987): Optimal local spline interpolants, J. Comput. Appl. Math.18, 107-119 · Zbl 0637.41009 · doi:10.1016/0377-0427(87)90059-8
[9] De Villiers, J.M., Rohwer, C.H. (1991): A nodal spline generalization of the Lagrange interpolant. In: P. Nevai, A. Pinkus, eds., Progress in approximation theory, pp. 201-211. Academic Press, San Diego
[10] De Villiers, J.M., Rohwer, C.H. (1992): Approximation properties of a nodal spline space (submitted for publication)
[11] Förster, K.-J. (1987): Über Monotonie und Fehlerkontrolle bei den Gregoryschen Quadraturverfahren, Z. Angew. Math. Mech.67, 257-266 · Zbl 0629.65022 · doi:10.1002/zamm.19870670612
[12] Lötzbeyer, W. (1972): Asymptotische Eigenschaften linearer und nichtlinearer Quadraturformeln, Z. Angew. Math. Mech.52, T211-T214 · Zbl 0245.65010
[13] Martensen, E. (1964): Optimale Fehlerschranken für die Quadraturformel von Gregory. Z. Angew. Math. Mech.44, 159-168 · Zbl 0119.33002 · doi:10.1002/zamm.19640440403
[14] Martensen, E. (1973): Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math.21, 70-80 · Zbl 0264.65021 · doi:10.1007/BF01436188
[15] Ralston, A., Rabinowitz, P. (1978): A First Course in Numerical Analysis. McGraw-Hill, New York · Zbl 0408.65001
[16] Schoenberg, I.J., Sharma, A. (1971): The interpolatory background of the Euler-Maclaurin quadrature formula, Bull. Amer. Math. Soc.77, 1034-1038 · Zbl 0271.41010 · doi:10.1090/S0002-9904-1971-12851-3
[17] Schumaker, L.L. (1981). Spline Functions: Basic Theory. Wiley, New York · Zbl 0449.41004
[18] Solak, W., Szydelko, Z. (1991): Quadrature rules with Gregory-Laplace end corrections. J. Comput. Appl. Math.36, 251-253 · Zbl 0738.65009 · doi:10.1016/0377-0427(91)90031-E
[19] Strang, G., Fix, G.J. (1973): An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, New Jersey · Zbl 0356.65096
[20] Stroud, A.H. (1966): Estimating quadrature errors for functions with low continuity. SIAM J. Numer. Anal.3, 420-424 · Zbl 0219.65027 · doi:10.1137/0703036
[21] Stroud, A.H. (1974): Numerical Quadrature and Solution of Ordinary Differential Equations. Appl. Math. Sci. 10, Springer, Berlin Heidelberg New York · Zbl 0298.65018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.