Numerische Mathematik

, Volume 66, Issue 1, pp 123–137 | Cite as

A nodal spline interpolant for the Gregory rule of even order

  • J. M. de Villiers


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.

Mathematics Subject Classification (1991)

41A55 41A15 41A05 65D32 65D30 65D07 65D05 


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  1. Atkinson, K.E. (1978): An Introduction to Numerical Analysis., Wiley, New YorkGoogle Scholar
  2. Babuška, I. (1966): Über die optimale Berechnung der Fourierschen Koeffizienten, Apl. Mat.11, 113–121Google Scholar
  3. Brass, H. (1977): Quadraturverfahren. Vandenhoeck & Ruprecht, GöttingenGoogle Scholar
  4. Dahmen, W., Goodman, T.N.T., Micchelli, C.A. (1988): Compactly supported fundamental functions for spline interpolation, Numer Math52, 639–664Google Scholar
  5. Davis, P.J., Rabinowitz, P. (1984): Methods of Numerical Integration, Academic Press, OrlandoGoogle Scholar
  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, OrlandoGoogle Scholar
  7. De Villiers, J.M. (1992): A convergence result in nodal spline interpolation, J. Approximation Theory (to appear)Google Scholar
  8. De Villiers, J.M., Rohwer, C.H. (1987): Optimal local spline interpolants, J. Comput. Appl. Math.18, 107–119Google Scholar
  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 DiegoGoogle Scholar
  10. De Villiers, J.M., Rohwer, C.H. (1992): Approximation properties of a nodal spline space (submitted for publication)Google Scholar
  11. Förster, K.-J. (1987): Über Monotonie und Fehlerkontrolle bei den Gregoryschen Quadraturverfahren, Z. Angew. Math. Mech.67, 257–266Google Scholar
  12. Lötzbeyer, W. (1972): Asymptotische Eigenschaften linearer und nichtlinearer Quadraturformeln, Z. Angew. Math. Mech.52, T211-T214Google Scholar
  13. Martensen, E. (1964): Optimale Fehlerschranken für die Quadraturformel von Gregory. Z. Angew. Math. Mech.44, 159–168Google Scholar
  14. Martensen, E. (1973): Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math.21, 70–80Google Scholar
  15. Ralston, A., Rabinowitz, P. (1978): A First Course in Numerical Analysis. McGraw-Hill, New YorkGoogle Scholar
  16. Schoenberg, I.J., Sharma, A. (1971): The interpolatory background of the Euler-Maclaurin quadrature formula, Bull. Amer. Math. Soc.77, 1034–1038Google Scholar
  17. Schumaker, L.L. (1981). Spline Functions: Basic Theory. Wiley, New YorkGoogle Scholar
  18. Solak, W., Szydelko, Z. (1991): Quadrature rules with Gregory-Laplace end corrections. J. Comput. Appl. Math.36, 251–253Google Scholar
  19. Strang, G., Fix, G.J. (1973): An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  20. Stroud, A.H. (1966): Estimating quadrature errors for functions with low continuity. SIAM J. Numer. Anal.3, 420–424Google Scholar
  21. Stroud, A.H. (1974): Numerical Quadrature and Solution of Ordinary Differential Equations. Appl. Math. Sci. 10, Springer, Berlin Heidelberg New YorkGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • J. M. de Villiers
    • 1
  1. 1.Department of MathematicsUniversity of StellenboschStellenboschSouth Africa

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