Skip to main content
SpringerLink
Log in

Construction of Symmetric Cubature Formulae with the Number of Knots (Almost) Equal to Möller’s Lower Bound

  • Chapter
Book cover Numerical Integration III

Abstract

We are concerned with determining the knots (x i ,y i ) and weights w i in a cubature formula

$$\sum\limits_{i = 1}^N {{w_i}} f\left( {{x_i},{y_i}} \right) $$

which is an approximation of

$$\iint\limits_R {w\left( {x,y} \right)f\left( {x,y} \right)dxdy.} $$

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Cools, R. and Haegemans, A. (1985) Construction of fully symmetric cubature formulae of degree 4k-3 for fully symmetric planar regions. Report TW71, Department of Computer Science, K.U.Leuven.

    Google Scholar 

  2. Cools, R. and Haegemans, A. (1987) Construction of fully symmetric cubature formulae of degree 4k-3 for fully symmetric planar regions. J. Comp. Appl. Math. 17, 173–180.

    Article  Google Scholar 

  3. Cools, R. (1984) Constructie van cubatuurformules voor tweedimensionale integralen met behulp van ideaaltheorie. Department of Computer Science, K.U.Leuven.

    Google Scholar 

  4. Cools, R. and Haegemans, A. (1987) Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller’s lower bound (with proofs). Report TW97, Department of Computer Science, K. U. Leuven.

    Google Scholar 

  5. Haegemans, A. and Piessens, R. (1976) Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials. Numer. Math. 25, 136–148.

    Google Scholar 

  6. Haegemans, A. and Piessens, R. (1977) Construction of cubature formulas of degree seven and nine for symmetric planar regions, using orthogonal polynomials. SIAM J. Numer. Anal. 14, 492–508.

    Article  Google Scholar 

  7. Macaulay, F.S. (1916) The algebraic theory of modular systems. Cambridge Tracts in Math. and Math. Physics no. 19, Cambridge Univ. Press.

    Google Scholar 

  8. Mantel, F. (1978) Rectangularly symmetric multidimensional cubature structures. WIS Technical Report, Weizmann Institute of Science.

    Google Scholar 

  9. Möller, H.M. (1973) Polynomideale und Kubaturformeln. Thesis, Univ. Dortmund.

    Google Scholar 

  10. Möller, H.M. and Mora, F. (1986) New constructive methods in classical ideal theory, Journal of Algebra 100, 138–178.

    Article  Google Scholar 

  11. Möller, H.M. (1987) On the construction of cubature formulae with few nodes using Gröbner bases. In: Keast P. and Fairweather, G.: Numerical Integration (Reidel Publishing Company, Dordrecht), 177–192.

    Chapter  Google Scholar 

  12. Morrow, C.R. and Patterson, T.N.L. (1978) Construction of algebraic cubature rules using polynomial ideal theory. SIAM J. Numer. Anal. 15, 953–976.

    Article  Google Scholar 

  13. Mysovskikh, I.P. (1968) On the construction of cubature formulas with fewest nodes. Soviet Math. Dokl. 9, 277–280.

    Google Scholar 

  14. Rabinowitz, P. and Richter, N. (1969) Perfectly symmetric two dimensional integration formulas with minimal numbers of points. Math. Comp. 23, 765–780.

    Article  Google Scholar 

  15. Schmid, H.J. (1979) Construction of cubature formulas using real ideals, In: Schempp, W. and Zeller, K.: Multivariate Approximation Theory (Birkhäuser Verlag, Stuttgart), 359–379.

    Chapter  Google Scholar 

  16. Stroud, A.H. (1971) Approximate calculation of multiple integrals. (Prentice Hall, Englewood Cliffs, N.Y.).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, B-3030, Heverlee, Belgium

    Ronald Cools & Ann Haegemans

Authors
  1. Ronald Cools
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ann Haegemans
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für angewandte Mathematik, TU Braunschweig, Pockelsstrasse 14, D-3300, Braunschweig, Germany

    H. Braß

  2. Mathematisches Institut, Ludwig-Maximilian-Universität, Theresienstrasse 39, D-8000, München 2, Germany

    G. Hämmerlin

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer Basel AG

About this chapter

Cite this chapter

Cools, R., Haegemans, A. (1988). Construction of Symmetric Cubature Formulae with the Number of Knots (Almost) Equal to Möller’s Lower Bound. In: Braß, H., Hämmerlin, G. (eds) Numerical Integration III. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 85. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6398-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-6398-8_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-2205-2

  • Online ISBN: 978-3-0348-6398-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation