No abstract

No abstract

No abstract

A procedure is reported for the compression of rank-deficient matrices. A matrix A of rank k is represented in the form A = U • B • V , where B is a k × k submatrix of A, and U , V are well-conditioned matrices that each contain a k × k identity submatrix. This property enables such compression schemes to be used in certain situations where the singular value decomposition (SVD) cannot be used efficiently. Numerical examples are presented. Introduction.In computational physics, and many other areas, one often encounters matrices whose ranks are (to high precision) much lower than their dimensionalities; even more frequently, one is confronted with matrices possessing large submatrices that are of low rank. An obvious source of such matrices is the potential theory, where discretization of integral equations almost always results in matrices of this type (see, for example, [7]). Such matrices are also encountered in fluid dynamics, numerical simulation of electromagnetic phenomena, structural mechanics, multivariate statistics, etc. In such cases, one is tempted to "compress" the matrices in question so that they could be efficiently applied to arbitrary vectors; compression also facilitates the storage and any other manipulation of such matrices that might be desirable.At this time, several classes of algorithms exist that use this observation. The so-called fast multipole methods (FMMs) are algorithms for the application of certain classes of matrices to arbitrary vectors; FMMs tend to be extremely efficient but are only applicable to very narrow classes of operators (see [2]). Another approach to the compression of operators is based on wavelets and related structures (see, for example, [3, 1]); these schemes exploit the smoothness of the elements of the matrix viewed as a function of their indices and tend to fail for highly oscillatory operators.Finally, there is a class of compression schemes that are based purely on linear algebra and are completely insensitive to the analytical origin of the operator. This class consists of the singular value decomposition (SVD), the so-called QR and QLP factorizations [10], and several others. Given an m×n matrix A of rank k < min(m, n), the SVD represents A in the formwhere D is a k × k diagonal matrix whose elements are nonnegative, and U and V are *

We consider the problem of analytically determining the effective thermal conductivity of a composite material consisting of periodic arrays of spheres with interfacial resistance. We applied Rayleigh's method which has been used extensively for such calculations in the perfect interface case, i.e. no jump in the temperature across the interface. Results are presented for simple, body-centred and face-centred cubic arrays, and each for a wide range of volume fractions. Our calculations were based on very accurate lattice sums in all three lattice cases and temperature fields were resolved to very high multipole moments, including all azimuthal terms. In the case of zero interfacial strength, our formulation recovers all previously reported benchmark results.