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Constructing cubature formulae: the science behind the art

Published online by Cambridge University Press:  07 November 2008

Ronald Cools
Ronald Cools
Affiliation:
Katholieke Universiteit LeuvenDept. of Computer Science Celestijnenlaan 200A B-3001 Heverlee, Belgium E-mail: Ronald.Cools@cs.kuleuven.ac.be
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Abstract

In this paper we present a general, theoretical foundation for the construction of cubature formulae to approximate multivariate integrals. The focus is on cubature formulae that are exact for certain vector spaces of polynomials. Our main quality criteria are the algebraic and trigonometric degrees. The constructions using ideal theory and invariant theory are outlined. The known lower bounds for the number of points are surveyed and characterizations of minimal cubature formulae are given. We include references to all known minimal cubature formulae. Finally, some methods to construct cubature formulae illustrate the previously introduced concepts and theorems.

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Type
Research Article
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Acta Numerica , Volume 6 , January 1997, pp. 1 - 54
Copyright
Copyright © Cambridge University Press 1997

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References

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