TY - JOUR

T1 - Commuting extensions and cubature formulae

AU - Degani, Ilan

AU - Schiff, Jeremy

AU - Tannor, David J.

PY - 2005/9

Y1 - 2005/9

N2 - Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to 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 briefly describe our attempts at computing them.

AB - Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to 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 briefly describe our attempts at computing them.

UR - http://www.scopus.com/inward/record.url?scp=29144519697&partnerID=8YFLogxK

U2 - 10.1007/s00211-005-0628-z

DO - 10.1007/s00211-005-0628-z

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AN - SCOPUS:29144519697

SN - 0029-599X

VL - 101

SP - 479

EP - 500

JO - Numerische Mathematik

JF - Numerische Mathematik

IS - 3

ER -