Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

Hladík, Milan

doi:10.1007/978-3-319-61753-4_11

Cited by 7 publications

(5 citation statements)

References 35 publications

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“…By Hladík [17], we have: This reduced the problem to checking positive definiteness of 2 K real matrices. Provided K is fixed, we arrived at a polynomial method for checking positive definiteness of A(p).…”

Section: Inverse M-matricesmentioning

confidence: 99%

An Overview of Polynomially Computable Characteristics of Special Interval Matrices

Hladík

2020

Studies in Computational Intelligence

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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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Section: Inverse M-matricesmentioning

confidence: 99%

An Overview of Polynomially Computable Characteristics of Special Interval Matrices

Hladík

2020

Studies in Computational Intelligence

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“…Positive definite and positive semidefinite interval matrices were studied, e.g., in [11,25,27]. An interval matrix A is strongly positive semidefinite if and only if the matrix A c −D s A ∆ D s ∈ A is positive semidefinite for each s ∈ {±1} n .…”

Section: Particular Matrix Classesmentioning

confidence: 99%

Stability of the linear complementarity problem properties under interval uncertainty

Hladík¹

2019

Preprint

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We consider the linear complementarity problem with uncertain data modeled by intervals, representing the range of possible values. Many properties of the linear complementarity problem (such as solvability, uniqueness, convexity, finite number of solutions etc.) are reflected by the properties of the constraint matrix. In order that the problem has desired properties even in the uncertain environment, we have to be able to check them for all possible realizations of interval data. This leads us to the robust properties of interval matrices. In particular, we will discuss S-matrix, Z-matrix, copositivity, semimonotonicity, column sufficiency, principal nondegeneracy, R 0 -matrix and R-matrix. We characterize the robust properties and also suggest efficiently recognizable subclasses.

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“…Considering positive definiteness, we have some sufficient conditions that can be checked polynomially [39]. In contrast to checking strong positive definiteness, weak positive definiteness can be checked in polynomial time by using semidefinite programming [15]; this polynomial result holds also for a more general class of symmetric interval matrices with linear dependencies [11]. For positive semidefiniteness it needn't be the case since semidefinite programming methods work only with some given accuracy.…”

Section: Positive Definitness and Semidefinitenessmentioning

confidence: 99%

Interval Linear Algebra and Computational Complexity

Horáček

Hladík

Černý³

2017

Springer Proceedings in Mathematics &Amp; Statistics

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This work connects two mathematical fields -computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebraregularity and singularity, full column rank, solving a linear system, deciding solvability of a linear system, computing inverse matrix, eigenvalues, checking positive (semi)definiteness or stability. We discuss these problems and relations between them from the view of computational complexity. Many problems in interval linear algebra are intractable, hence we emphasize subclasses of these problems that are easily solvable or decidable. The aim of this work is to provide a basic insight into this field and to provide materials for further reading and research. IntroductionThe purpose of this work is to emphasize relations between the two mathematical fields -interval linear algebra and computational complexity. This is not a pioneer work. Variety of relations between interval problems and computational complexity is covered by many papers. There are also few monographs that are devoted to this topic [4,22,46]. Some questions may arise in mind while reading the previous works. Among all, it is the question about the equivalence of the notions NPhardness and co-NP-hardness. Some authors use these notions as synonyms. Some distinguish between them. Another questions that may arise touches the representation and reducibility of interval problems in a given computational model. We would like to shed more light (not only) on these issues.Many well-known problems of classical linear algebra become intractable when we introduce intervals into matrices and vectors. However, not everything is lost. There are many interesting sub-classes of problems that behave well. We would like to point out these feasible cases, since they are interesting either from the theoretical or the computational point of view.Our work does not aspire to replace the classical monographs or handbooks. It lacks many of their details that are cited in the text. Nevertheless, it collects even some recent results that are missing in the monographs. It also provides links and reductions between the various areas of interval linear algebra. It provides a necessary and compact introduction to computational complexity and interval linear algebra. Then it considers complexity and feasibility of various well-known linear algebraic tasks when considered with interval structures -regularity and singularity, full column rank, solving a linear system, deciding solvability of a linear system, computing inverse matrix, eigenvalues, checking positive (semi)definiteness or stability.We hope this paper should help newcomers to this area to improve her/his orientation in the field or professionals to provide a signpost to more deeper literature. Interval linear algebra -part IInterval linear algebra is a mathematical field developed from classical linear algebra. The only difference is, that we do not work with real numbers but with real closed intervalswhere a ≤ a. The set of all closed real intervals i...

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Positive Semidefiniteness and Positive Definiteness of a Linear Parametric Interval Matrix

Cited by 7 publications

References 35 publications

An Overview of Polynomially Computable Characteristics of Special Interval Matrices

An Overview of Polynomially Computable Characteristics of Special Interval Matrices

Stability of the linear complementarity problem properties under interval uncertainty

Interval Linear Algebra and Computational Complexity

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