4,190 research outputs found
On the nearest singular matrix pencil
Given a regular matrix pencil A + μE, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil A + ΔA + μ(E + ΔE), one approach for general singular pencils and another one such that A+ ΔA and E + ΔE have a common left/right null vector. For the latter case the nearest singular pencil is shown to differ from the original pencil by rank-one matrices ΔA and ΔE. In both cases we consider also the situation where only A is perturbed. The nearest singular pencil is approached by a two-level iteration, where a gradient flow is driven to a stationary point in the inner iteration and the outer level uses a fast iteration for the distance parameter. This approach extends also to structured matrices A and E
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
Verified partial eigenvalue computations using contour integrals for Hermitian generalized eigenproblems
We propose a verified computation method for partial eigenvalues of a
Hermitian generalized eigenproblem. The block Sakurai-Sugiura Hankel method, a
contour integral-type eigensolver, can reduce a given eigenproblem into a
generalized eigenproblem of block Hankel matrices whose entries consist of
complex moments. In this study, we evaluate all errors in computing the complex
moments. We derive a truncation error bound of the quadrature. Then, we take
numerical errors of the quadrature into account and rigorously enclose the
entries of the block Hankel matrices. Each quadrature point gives rise to a
linear system, and its structure enables us to develop an efficient technique
to verify the approximate solution. Numerical experiments show that the
proposed method outperforms a standard method and infer that the proposed
method is potentially efficient in parallel.Comment: 15 pages, 4 figures, 1 tabl
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
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