An Overview of Polynomially Computable Characteristics of Special Interval Matrices

Hladík, Milan

doi:10.1007/978-3-030-31041-7_16

Cited by 7 publications

(4 citation statements)

References 39 publications

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“…In the next proposition, we give a formula for symmetric matrices. Before we get to the proposition, we present a lemma, which follows directly from [21,Thm. 17].…”

Section: Condition Number Of Ave For 2-normmentioning

confidence: 99%

Error bounds and a condition number for the absolute value equations

Zamani

Milan

2022

Math. Program.

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Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations.

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“…In the next proposition, we give a formula for symmetric matrices. Before we get to the proposition, we present a lemma, which follows directly from [21,Thm. 17].…”

Section: Condition Number Of Ave For 2-normmentioning

confidence: 99%

Error bounds and a condition number for the absolute value equations

Zamani

Milan

2022

Math. Program.

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“…Let A be a positive definite M-matrix. By [13], the interval matrix A = [A − E, A + E] is positive definite if and only if it is an H-matrix. Thus, the radius of positive definiteness of this A can be handled by techniques from Section 6.…”

Section: Proposition 1 If a Is Inverse Nonnegative Then For The Max-n...mentioning

confidence: 99%

Tolerances, robustness and parametrization of matrix properties related to optimization problems

Hladík

2018

Optimization

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When we speak about parametric programming, sensitivity analysis, or related topics, we usually mean the problem of studying specified perturbations of the data such that for a given optimization problem some optimality criterion remains satisfied. In this paper, we turn to another question. Suppose that A is a matrix having a specific property P. What are the maximal allowable variations of the data such that the property still remains valid for the matrix? We study two basic forms of perturbations. The first is a perturbation in a given direction, which is closely related to parametric programming. The second type consists of all possible data variations in a neighbourhood specified by a certain matrix norm; this is related to the tolerance approach to sensitivity analysis, or to stability. The matrix properties discussed in this paper are positive definiteness; P-matrix, H-matrix and P-matrix property; total positivity; inverse M-matrix property and inverse nonnegativity.

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“…This class also includes interval M-matrices [3], inverse nonnegative [21] or totally positive matrices [6] as particular subclasses that are efficiently recognizable; cf. [13]. 5.6.…”

Section: Monotonicity Checking the Derivative Of A Real Nonsingular mentioning

confidence: 99%

Determinants of Interval Matrices

Horáček

Hladík

Matějka

2018

ELA

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In this paper we shed more light on determinants of interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determinants and their possible combination with preconditioning are discussed. A new method based on Cramer's rule was designed. It returns similar results to the state-of-the-art method, however, it is less consuming regarding computational time. As a byproduct, the Gerschgorin circles were generalized for interval matrices. New results about classes of interval matrices with polynomially computable tasks related to determinant are proved (symmetric positive definite matrices, class of matrices with identity midpoint matrix, tridiagonal H-matrices). The mentioned methods were exhaustively compared for random general and symmetric matrices.

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An Overview of Polynomially Computable Characteristics of Special Interval Matrices

Cited by 7 publications

References 39 publications

Error bounds and a condition number for the absolute value equations

Error bounds and a condition number for the absolute value equations

Tolerances, robustness and parametrization of matrix properties related to optimization problems

Determinants of Interval Matrices

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