Kubaturformeln mit minimaler Knotenzahl

Möller, H. Michael

doi:10.1007/bf01462272

Cited by 132 publications

(81 citation statements)

References 4 publications

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“…The key point consists clearly in finding suitable algebraic cubature formulas as in (3) for the product Chebyshev measure, with a low number of nodes. We recall that the number of nodes of a minimal formula with degree of exactness 2n+1, according to a general result by Möller [26] on centrally symmetric weight functions, is…”

Section: Nontensorial Clenshaw-curtis Cubaturementioning

confidence: 99%

Nontensorial Clenshaw–Curtis cubature

2008

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We extend Clenshaw-Curtis quadrature to the square in a nontensorial way, by using Sloan's hyperinterpolation theory and two families of points recently studied in the framework of bivariate (hyper)interpolation, namely the Morrow-Patterson-Xu points and the Padua points. The construction is an application of a general approach to product-type cubature, where we prove also a relevant stability theorem. The resulting cubature formulas turn out to be competitive on nonentire integrands with tensorproduct Clenshaw-Curtis and Gauss-Legendre formulas, and even with the few known minimal formulas.2000 AMS subject classification: 65D32.

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Section: Nontensorial Clenshaw-curtis Cubaturementioning

confidence: 99%

Nontensorial Clenshaw–Curtis cubature

2008

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“…Then • Notes: (1) As mentioned in the introduction, theorem 12 has its origins in the work of Möller [6]. A statement of the result in a form that clearly corresponds to our statement can be found in [17], which cites [18] and [19].…”

Section: Theorem 12mentioning

confidence: 57%

“…Subsection 5.3 includes some simple consequences of this relationship. One of the key results in the theory of cubature rules, a lower bound on the number of nodes needed for an odd degree cubature rule, originating in the work of Möller [6], follows from a general result in the theory of commuting extensions (theorem 2 in section 3). Similarly a simple result on the spectra of commuting extensions (theorem 6 in section 3) gives interesting constraints on the nodes in positive weight cubature rules, which we believe have hitherto been overlooked even in one dimension.…”

Section: Introductionmentioning

confidence: 99%

Commuting extensions and cubature formulae

2005

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Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A 1 , . . . , A d , related to the coordinate operators x 1 , . . . , x d , in R d . We prove a correspondence between cubature formulae and "commuting extensions" of A 1 , . . . , A d , satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus, the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.

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“…On the other hand, a minimal formula of degree 3 has four nodes; it follows from the general theory of minimal cubature formulae [14] that these nodes are common zeros of two orthogonal polynomials of degree 2. Since a basis of the orthogonal polynomials with respect to WQ is known explicitly (cf.…”

Section: /mentioning

confidence: 99%

“…We found the nodes and the weights with the help of the computer program Mathematica. For more on the minimal cubature formula, we refer the reader to [14,15,21,22] and the references there. Since all nodes of (4.11) are located inside the triangle E 2 , it yields a Z2 x Z2 x Z2 invariant cubature formula of degree 7 with 32 nodes, 4 /(Xi, X2, Xs)^ = -^ Ai ^ /( db y/s~U ±y/t u ±yjl -S; -U ) (4.12)…”

Section: /mentioning

confidence: 99%

Orthogonal polynomials and cubature formulae on spheres and on simplices

Xu¹

1998

Methods and Applications of Analysis

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Kubaturformeln mit minimaler Knotenzahl

Cited by 132 publications

References 4 publications

Nontensorial Clenshaw–Curtis cubature

Nontensorial Clenshaw–Curtis cubature

Commuting extensions and cubature formulae

Orthogonal polynomials and cubature formulae on spheres and on simplices

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