303 research outputs found
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
Subsquares Approach - Simple Scheme for Solving Overdetermined Interval Linear Systems
In this work we present a new simple but efficient scheme - Subsquares
approach - for development of algorithms for enclosing the solution set of
overdetermined interval linear systems. We are going to show two algorithms
based on this scheme and discuss their features. We start with a simple
algorithm as a motivation, then we continue with a sequential algorithm. Both
algorithms can be easily parallelized. The features of both algorithms will be
discussed and numerically tested.Comment: submitted to PPAM 201
Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix
We consider a symmetric matrix, the entries of which depend linearly on some
parameters. The domains of the parameters are compact real intervals. We
investigate the problem of checking whether for each (or some) setting of the
parameters, the matrix is positive definite (or positive semidefinite). We
state a characterization in the form of equivalent conditions, and also propose
some computationally cheap sufficient\,/\,necessary conditions. Our results
extend the classical results on positive (semi-)definiteness of interval
matrices. They may be useful for checking convexity or non-convexity in global
optimization methods based on branch and bound framework and using interval
techniques
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