On the Smolyak cubature error for analytic functions

Advances in Computational Mathematics

Volumn 12, Issue 1, 2000, Pages 71-93

  Petras, Knut ^a  

^a TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

Author keywords

Analytic functions; Clenshaw Curtis quadrature; Cubature; Error bounds; Kronrod Patterson quadrature; Smolyak algorithm

Indexed keywords

EID: 0034405305     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1018904816230     Document Type: Article

References (17)

1. N.L. Achieser, Vorlesungen über Appoximationstheorie (Akademie-Verlag, Berlin, 1967) (German translation).
2. H. Brass, Quadraturverfahren (Vandenhoeck & Ruprecht, Göttingen, 1977).
3. R.E. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numer. 7 (1998) 1-49.
4. T. Gerstner and M. Griebel, Numerical integration using sparse grids, Numer. Algorithms 18 (1998) 209-232.
5. I.S. Gradshteyn and I.M. Ryzhik, Table of Series, Integrals and Products (Academic Press, New York, 1965).
6. E.R. Hansen, A Table of Series and Products (Prentice-Hall, Englewood Cliffs, NJ, 1975).
7. R. Klatte, U. Kulisch, M. Neaga, D. Ratz and C. Ullrich, PASCAL-XSC. Language Reference with Examples (Springer, Berlin, 1992).
8. W. Krämer, U. Kulisch and R. Lohner, Numerical Toolbox for Verified Computing II, in preparation for Springer, Berlin.
9. F.G. Lether, Some observations on Gauss-Legendre quadrature error estimates for analytic functions, J. Comput. Appl. Math. 7 (1981) 63-66.
10. G. Monegato, Stieltjes polynomials and related quadrature rules, SIAM Rev. 24 (1982) 137-158.
11. G. Neumann, Boolesche interpolatorische Kubatur, Dissertation, Universität Gesamthochschule Siegen (1982).
12. E. Novak and K. Ritter, High-dimensional integration of smooth functions over cubes, Numer. Math. 75 (1996) 79-97.
13. T.N.L. Patterson, The optimum addition of points to quadrature formulae, Math. Comp. 22 (1968) 847-856.
14. K. Petras, Gaussian integration of Chebyshev polynomials and analytic functions, Numer. Algorithms 10 (1995) 187-202.
15. I.H. Sloan and S. Joe, Lattice Methods for Multiple Integration (Clarendon Press, Oxford, 1994).
16. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl. 4 (1963) 240-243.
17. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, CA, 1991).