Regularity radius and real eigenvalue range

Kolev, Lubomir V.

doi:10.1016/j.amc.2014.01.113

Cited by 3 publications

(4 citation statements)

References 13 publications

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“…Radius of regularity for interval parametric matrices was first defined in [24] under the assumption that r * (A(p), p) < ∞. In several works, e.g., [10], [24], L. Kolev develops and applies a methodology for either finding the regularity radius of an interval parametric matrix or providing bounds for the latter. Beside various other requirements, this methodology also assumes r * (A(p), p) < ∞.…”

Section: Radius Of Regularitymentioning

confidence: 99%

Necessary and sufficient conditions for regularity of interval parametric matrices

Popova

2021

Preprint

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Matrix regularity is a key to various problems in applied mathematics. The sufficient conditions, usually used for checking regularity of interval parametric matrices, fail in case of large parameter intervals. We present necessary and sufficient conditions for regularity of interval parametric matrices in terms of boundary parametric hypersurfaces, parametric solution sets, determinants, real spectral radiuses. The initial n-dimensional problem involving K interval parameters is replaced by numerous problems involving 1 ≤ t ≤ min{n − 1, K} interval parameters, in particular t = 1 is most attractive. The advantages of the proposed methodology are discussed along with its application for finding the interval hull solution to interval parametric linear system and for determining the regularity radius of an interval parametric matrix.

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Section: Radius Of Regularitymentioning

confidence: 99%

Necessary and sufficient conditions for regularity of interval parametric matrices

Popova

2021

Preprint

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“…Whereas sometimes, as pointed by Kolev [6], it might be useful to have a quantity, which would measure the distance of a parametric matrix from singularity. The radius of regularity of a normalised parametric interval matrix with affine-linear dependencies, defined as ( [6])…”

Section: Regularitymentioning

confidence: 99%

“…can be considered as such measure. Kolev [6] proposed an explicit formula for r(A(p)), which can be useful to develop an algorithm for computing the radius of regularity, which is not a priori exponential.…”

Section: Regularitymentioning

confidence: 99%

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Positive definiteness and stability of parametric interval matrices

Skalna¹

2017

Preprint

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We investigate positive definiteness, Hurwitz stability and Schur stability of parametric interval matrices. We give a verifiable sufficient condition for positive definiteness of parametric interval matrices with non-linear dependencies. We also give several sufficient and necessary conditions for stability of symmetric parametric interval matrices with affine-linear dependencies. The presented results extend the results on positive definiteness and stability of interval matrices. In addition, we provide a formula for the radius of stability of symmetric parametric interval matrices.

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Regularity Radius: Properties, Approximation and a Not a Priori Exponential Algorithm

Hartman

Hladík

2018

ELA

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The radius of regularity sometimes spelled as the radius of nonsingularity is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. There are basically two approaches to handle this situation. Firstly, approximation algorithms are applied and secondly, tighter bounds for radius of regularity are applied. Improvements of both approaches have been recently shown by Hartman and Hladík (doi:10.1007/978-3-319-31769-4 9) utilizing relaxation of computation to semidefinite programming. An estimation of the regularity radius using either of mentioned ways is usually applied to general matrices considering none or just weak assumptions about the original matrix. Surprisingly less explored area is represented by utilization of properties of special class of matrices as well as utilization of classical algorithms extended to be used to compute the considered radius. This work explores a process of regularity radius analysis and identifies useful properties enabling easier estimation of the corresponding radius values. At first, checking finiteness of this characteristic is shown to be a polynomial problem along with determining a maximal bound on number of nonzero elements of the matrix to obtain infinite radius. Further, relationship between maximum (Chebyshev) norm and spectral norm is used to construct new bounds for the radius of regularity. Considering situations where known bounds are not enough, a new method based on Jansson-Rohn algorithm for testing regularity of an interval matrix is presented which is not a priory exponential along with numerical experiments. For a situation where an input matrix has a special form, several corresponding results are provided such as exact formulas for several special classes of matrices, e.g., for totally positive and inverse non-negative, or approximation algorithms, e.g., rank-one radius matrices. For tridiagonal matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design a polynomial algorithm to compute the radius of regularity.

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Regularity radius and real eigenvalue range

Cited by 3 publications

References 13 publications

Necessary and sufficient conditions for regularity of interval parametric matrices

Necessary and sufficient conditions for regularity of interval parametric matrices

Positive definiteness and stability of parametric interval matrices

Regularity Radius: Properties, Approximation and a Not a Priori Exponential Algorithm

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