Computing

, Volume 62, Issue 2, pp 147–162

Smolyak's Construction of Cubature Formulas of Arbitrary Trigonometric Degree

  • R. Cools
  • E. Novak
  • K. Ritter

DOI: 10.1007/s006070050018

Cite this article as:
Cools, R., Novak, E. & Ritter, K. Computing (1999) 62: 147. doi:10.1007/s006070050018

Abstract.

We study cubature formulas for d-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degree \(\ell\) in dimension d, we need about \(d^\ell/\ell !\) function values if d is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare different methods to obtain formulas with high trigonometric degree.

AMS Subject Classifications:Primary 41A55, 41A63; Secondary 65D32.
Key words.Cubature formulas, trigonometric degree, Smoyak construction.

Copyright information

© Springer Verlag, Wien, 1999

Authors and Affiliations

  • R. Cools
    • 1
  • E. Novak
    • 2
  • K. Ritter
    • 3
  1. 1.Department of Computer Science Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium, e-mail: ronald.cools@cs.kuleuven.ac.be BE
  2. 2.Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstraße 1 1/2, D-91054 Erlangen, Germany, e-mail: novak@mi.uni-erlangen.de DE
  3. 3.Fakultät für Mathematik und Informatik, Universität Passau, D-94030 Passau, Germany, e-mail: klaus.ritter@fmi.uni-passau.de DE