On Relation Between P-Matrices and Regularity of Interval Matrices (original) (raw)
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We present a sufficient regularity condition for interval matrices which generalizes two previously known ones. It is formulated in terms of positive definiteness of a certain point matrix, and can also be used for checking positive definiteness of interval matrices. Comparing it with Beeck's strong regularity condition, we show by counterexamples that none of the two conditions is more general than the other one. Above: logo of interval computations and related areas (depiction of the solution set of the system [2, 4]x 1 + [−2, 1]x 2 = [−2, 2], [−1, 2]x 1 + [2, 4]x 2 = [−2, 2] (Barth and Nuding [1])).
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AE Regularity of Interval Matrices
The Electronic Journal of Linear Algebra, 2018
Consider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by ââ- quantification. The paper deals with the problem of what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, a concept of AE regularity is introduced, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. A characterization of AE regularity is discussed, and also various classes of matrices that are implicitly AE regular are investigated. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. Eventually, there are also stated open problems related to computational complexity and characterization of AE regularity.
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It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some of such classes. In particular, we focus on several special interval matrices and investigate their convenient properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse M-matrices, inverse nonnegative matrices, nonnegative matrices, totally positive matrices and some others. We focus in particular on computing the range of the determinant, eigenvalues, singular values, and selected norms. Whenever possible, we state also formulae for determining the inverse matrix and the hull of the solution set of an interval system of linear equations. We survey not only the known facts, but we present some new views as well.
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For interval matrices, the paper considers the problem of determining whether a matrix has full rank. We propose a full-rank criterion that relies on the search for diagonal dominance as well as criteria based on pseudoinversion of the midpoint matrix and comparison of the midpoint and the radius matrices for the interval matrix under study.
Diagonal stability of interval matrices and applications
Linear Algebra and its Applications, 2010
P is a positive definite diagonal matrix and the notation "≺ 0" means negative definite. The first part of the paper • provides SDS p and HDS p criteria, • presents methods for finding the positive definite diagonal matrix requested by the definition of SDS p and HDS p , • analyzes the robustness of SDS p and HDS p and • explores the connection with the Schur and Hurwitz stability of A I . The second part shows that the SDS p or HDS p of A I is equivalent to the following properties of a discrete-or continuous-time dynamical interval system whose motion is described by A I : • the existence of a strong Lyapunov function defined by the p-norm and • the existence of exponentially decreasing sets defined by the p-norm that are invariant with respect to system's trajectories.
Interval matrix norms induced by point matrix norms are introduced in the space of interval matrices. It is shown that evaluating the interval matrix norm induced by a point matrix norm · p is exponential (probably NP-hard) for p = 2 and requires computation of only one point matrix norm for p ∈ {1, ∞, (1, ∞), F }. . 4 Above: logo of interval computations and related areas (depiction of the solution set of the system [2, 4]x 1 + [−2, 1]x 2 = [−2, 2], [−1, 2]x 1 + [2, 4]x 2 = [−2, 2] (Barth and Nuding [3])).
Improved bounds for the spectrum of interval matrices
IET Control Theory & Applications, 2013
This study presents new sufficient conditions for Hurwitz and Schur stability of interval matrices. Tight bounds for the spectrum of interval matrices are estimated using computationally simple optimisation problems. The conservativeness is reduced further by application of ordinary similarity transformation. A necessary and sufficient vertex based criterion for the stability of a subclass of interval systems in continuous and discrete-time cases is also proposed. This enables the spectra for this class of interval systems to be determined exactly. A selection of various examples adopted from existing literature is used to demonstrate the utility of the proposed criteria.
Matrices with the consecutive ones property, interval graphs and their applications
2001
Matrices with the consecutive ones property and interval graphs are important notations in the field of applied mathematics. We give a theoretical picture of them in first part. We present the earliest work in interval graphs and matrices with the consecutive ones property pointing out the close relation between them. We pay attention to Tucker's structure theorem on matrices with the consecutive ones property as an essential step that requires a deep considerations. Later on we concentrate on some recent work characterizing the matrices with the consecutive ones property and matrices related to them in the terms of interval digraphs as the latest and most interesting outlook on our topic. Within this framework we introduce a classiffcation of matrices with consecutive ones property and matrices related to them. We describe the applications of matrices with the consecutive ones property and interval graphs in different fields. We make sure to give a general view of application a...