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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Cubature formulas for symmetric measures in higher dimensions with few points


Authors: Aicke Hinrichs and Erich Novak
Journal: Math. Comp. 76 (2007), 1357-1372
MSC (2000): Primary 65D32
Published electronically: February 16, 2007
MathSciNet review: 2299778
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Abstract | References | Similar Articles | Additional Information

Abstract: We study cubature formulas for $ d$-dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree $ \ell=5$ or $ \ell=7$ and large dimension $ d$ the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions.

We also show, for any odd degree $ \ell = 2k+1$, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.


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Additional Information

Aicke Hinrichs
Affiliation: Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Email: hinrichs@math.uni-jena.de

Erich Novak
Affiliation: Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Email: novak@math.uni-jena.de

DOI: http://dx.doi.org/10.1090/S0025-5718-07-01974-6
PII: S 0025-5718(07)01974-6
Keywords: Cubature formulas, M\"oller bound, Smolyak method, polynomial exactness
Received by editor(s): August 25, 2005
Received by editor(s) in revised form: June 16, 2006
Published electronically: February 16, 2007
Additional Notes: Research of the first author was supported by the DFG Emmy-Noether grant Hi 584/2-4.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.