605 research outputs found
On the decay of the inverse of matrices that are sum of Kronecker products
Decay patterns of matrix inverses have recently attracted considerable
interest, due to their relevance in numerical analysis, and in applications
requiring matrix function approximations. In this paper we analyze the decay
pattern of the inverse of banded matrices in the form where is tridiagonal, symmetric and positive definite, is
the identity matrix, and stands for the Kronecker product. It is well
known that the inverses of banded matrices exhibit an exponential decay pattern
away from the main diagonal. However, the entries in show a
non-monotonic decay, which is not caught by classical bounds. By using an
alternative expression for , we derive computable upper bounds that
closely capture the actual behavior of its entries. We also show that similar
estimates can be obtained when has a larger bandwidth, or when the sum of
Kronecker products involves two different matrices. Numerical experiments
illustrating the new bounds are also reported
Flux vector splitting of the inviscid equations with application to finite difference methods
The conservation-law form of the inviscid gasdynamic equations has the remarkable property that the nonlinear flux vectors are homogeneous functions of degree one. This property readily permits the splitting of flux vectors into subvectors by similarity transformations so that each subvector has associated with it a specified eigenvalue spectrum. As a consequence of flux vector splitting, new explicit and implicit dissipative finite-difference schemes are developed for first-order hyperbolic systems of equations. Appropriate one-sided spatial differences for each split flux vector are used throughout the computational field even if the flow is locally subsonic. The results of some preliminary numerical computations are included
Minimizing Communication for Eigenproblems and the Singular Value Decomposition
Algorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
-like algorithms for linear algebra, including eigenvalue problems and
the SVD. Conventional algorithms, including those currently implemented in
(Sca)LAPACK, perform asymptotically more communication than these lower bounds
require. In this paper we present parallel and sequential eigenvalue algorithms
(for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms
that do attain these lower bounds, and analyze their convergence and
communication costs.Comment: 43 pages, 11 figure
Representation of the three-body Coulomb Green's function in parabolic coordinates: paths of integration
The possibility is discussed of using straight-line paths of integration in
computing the integral representation of the three-body Coulomb Green's
function. In our numerical examples two different integration contours are
considered. It is demonstrated that only one of these straight-line paths
provides that the integral representation is valid
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