Abstract
We study a new simple quadrature rule based on integrating a C 1 quadratic spline quasi-interpolant on a bounded interval. We give nodes and weights for uniform and non-uniform partitions. We also give error estimates for smooth functions and we show that our formula is a useful companion to Simpson’s rule.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. de Boor, A Practical Guide to Splines, revised edn., Springer, New York, 2001.
P.J. Davis, Interpolation and Approximation, Dover Publ., New York, 1975.
P. J. Davis and P. Rabinowitz, Numerical Integration, 2nd edn., Academic Press, London-New York, 1984.
S. A. De Swardt and J. M. De Villiers, Gregory type quadrature based on quadratic nodal spline interpolation, Numer. Math., 85 (2000), pp. 129–153.
R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin-New York, 1993.
H. Engels, Numerical Quadrature and Cubature, Academic Press, London-New York, 1980.
W. Gautschi, Orthogonal polynomials: applications and computation, in Acta Numer., A. Iserles (ed.), vol. 5, Cambridge University Press, Cambridge, 1996, pp. 45–119.
W. Gautschi, Orthogonal Polynomials. Cambridge University Press, 2004.
W. J. Kammerer, G. W. Reddien, and R. S. Varga, Quadratic interpolatory splines, Numer. Math., 22 (1974), pp. 241–259.
A. Krommer and C. W. Ueberhuber, Computational Integration, SIAM, Philadelphia, 1998.
V. Lampret, An invitation to Hermite’s integration and summation: a comparison between Hermite’s and Simpson’s rules, SIAM Rev., 46(2) (2004), pp. 329–345.
J. M. Marsden, An identity for spline functions with applications to variation diminishing spline approximation, J. Approx. Theory, 3 (1970), pp. 7–49.
J. M. Marsden, Operator norm bounds and error bounds for quadratic spline interpolation, in Approximation Theory, Banach Center Publications, vol. 4, Polish Scientific Publishers, Warsaw, 1979, pp. 159–175.
M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.
P. Sablonnière, On some multivariate quadratic spline quasi-interpolants on bounded domains, in Modern Developments in Multivariate Approximation, W. Haussmann et al. (eds.), ISNM, Int. Ser. Numer. Math., vol. 145, Birkhäuser, Basel, 2003, pp. 263–278.
P. Sablonnière, Quadratic spline quasi-interpolants on bounded domains of ℝd, d=1,2,3, Rend. Sem. Univ. Pol. Torino, 61 (2003), pp. 61–78.
P. Sablonnière, Spline quasi-interpolants and applications to numerical analysis, Rend. Sem. Univ. Pol. Torino, 63(2) (2005), pp. 107–118.
P. Sablonnière, Recent progress on univariate and multivariate polynomial and spline quasi-interpolants, in Trends and Applications in Constructive Approximation, M.G. de Bruijn et al. (eds), ISNM, Int. Ser. Numer. Math., vol. 151, Birkhäuser, Basel, 2005, pp. 229–245.
L. L. Schumaker, Spline Functions: Basic Theory, John Wiley & Sons, New York, 1981.
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS subject classification (2000)
41A15, 65D07, 65D25, 65D32
Rights and permissions
About this article
Cite this article
Sablonnière, P. A quadrature formula associated with a univariate spline quasi interpolant . Bit Numer Math 47, 825–837 (2007). https://doi.org/10.1007/s10543-007-0146-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10543-007-0146-8