Abstract
The knots and weights of a cubature formula are determined by a system of nonlinear equations. The number of equations and unknowns can be reduced by imposing some structure on the formula.
We are concerned with the construction of cubature formulae which are invariant under rotations. Using invariant theory, we obtain a smaller system of algebraically independent equations.
This is used to construct cubature formulae for the square. One of the results is a 24-point formula of degree 11.
Zusammenfassung
Die Knoten und Gewichte einer Kubaturformel sind bestimmt durch ein nichtlineares Gleichungssystem Die Zahl der Gleichungen und Unbekannten kann man verringern, indem man der Formel eine gewisse Struktur auferlegt.
Wir bestimmen rotationsinvariante Kubaturformeln. Die Invarianztheorie liefert ein kleiners System algebraisch unabhängiger Gleichungen.
Damit konstruieren wir Kubaturformeln für das Quadrat. Ein Ergebnis ist eine 24-Punkt-Formel vom Grad 11.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Beckers, M., Haegemans, A.: Construction of three-dimensional invariant cubature formulae. Report TW85. K. U. Leuven, Department of Computer Science 1986.
Burnside, W.: Theory of Groups of Finite Orders. Dover: New York 1955.
Cools, R., Haegemans, A.: Construction of minimal cubature formulae for the square and the triangle, using invariant theory. Report TW96, K. U. Leuven, Department of Computer Science 1987.
Cools, R., Haegemans, A.: Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller's lower bound. Report TW97, K. U. Leuven, Department of Computer Science 1987.
Cools, R., Haegemans, A.: Automatic computation of knots and weights of cubature formulae for circular symmetric planar regions. J. Comp. Appl. Math.20, 153–158 (1987).
Haegemans, A., Cools, R.: Construction of three-dimensional cubature formulae with points on regular and semi-regular polytopes. In: Numerical Integration, Recent Developments, Software and Applications (Keast, P., Fairweather, G., eds.), pp. 153–163. Dordrecht: Reidel 1986.
Haegemans, A., Piessens, R.: Construction of cubature formulas of degree seven and nine for symmetric planar regions, using orthogonal polynomials. SIAM J. Numer. Anal.14, 492–508 (1977).
Jacobson, N.: Lectures in Abstract Algebra. Vol. 3. Princeton, N. J.: Van Nostrand 1964.
Mantel, F.: Personal communication (1986).
Molien, T.: Über die Invarianten der linearen Substitutionsgruppen. Berliner Sitzungsberichte, pp. 1152–1156 (1898).
Möller, H. M.: Kubaturformeln mit minimaler Knotenzahl. Numer. Math.25, 185–200 (1976).
Möller, H. M., Schmid, H. J.: Problem 7, Kubaturformeln des Grads 11 mit 24 Knoten. In: Numerical Integration (Hammerlin, G., ed.), pp. 273. Birkhäuser 1982.
Rabinowitz, P., Richter, N.: Perfectly symmetric two-dimensional integration formulas with minimal number of points. Math. Comp.23, 765–779 (1969).
Sloane, N. J. A.: Error-correcting codes and invariant theory: new applications of a nineteenthcentury technique. Amer. Math. Monthly84, 82–107 (1977).
Sobolev, S. L.: Formulas of mechanical cubature on the surface of a sphere (in Russ.). Sibirsk. Mat. Z.3, 769–796 (1962).
Stroud, A. H.: Approximate Calculation of Multiple Integrals. Englewood Cliffs, N. J.: Prentice-Hall 1971.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cools, R., Haegemans, A. Another step forward in searching for cubature formulae with a minimal number of knots for the square. Computing 40, 139–146 (1988). https://doi.org/10.1007/BF02247942
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02247942