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27
Computing exact componentwise bounds on solution of linear systems with interval data is NPhard
 SIAM J. Matrix Anal. Appl
, 1995
"... Abstract. We prove that it is NPhard to compute the exact componentwise bounds on solutions of all the linear systems which can be obtained from a given linear system with a nonsingular matrix by perturbing all the data independently of each other within prescribed tolerances. (1) ..."
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Abstract. We prove that it is NPhard to compute the exact componentwise bounds on solutions of all the linear systems which can be obtained from a given linear system with a nonsingular matrix by perturbing all the data independently of each other within prescribed tolerances. (1)
Analysis of Linear Mechanical Structures With Uncertainties By Means of Interval Methods
, 1998
"... ... of this paper is to investigate possibilities of and problems with application of interval methods in (qualitative) analysis of linear mechanical systems with parameter uncertainties, in particular truss structures and frames. The paper starts with an introduction to interval arithmetic and sy ..."
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Cited by 7 (3 self)
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... of this paper is to investigate possibilities of and problems with application of interval methods in (qualitative) analysis of linear mechanical systems with parameter uncertainties, in particular truss structures and frames. The paper starts with an introduction to interval arithmetic and systems of linear interval equations, including an overview of basic methods for finding interval estimates for the set of solutions of such systems. The methods are further illustrated by several examples of practical problems, solved by our hybrid system of analysis of mechanical structures. Finally, several general problems with using interval methods for analysis of such linear systems are identified, with promising avenues for further research indicated as a result. The problems discussed include estimation inaccuracy of the algorithms (especially the fundamental problem of matrix coefficient dependence), their computational complexity, as well as inadequate development of methods for analysis of interval systems with singular matrices.
On the Complexity of Matrix Rank and Rigidity
"... We revisit a well studied linear algebraic problem, computing the rank and determinant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for L. We show that computi ..."
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Cited by 5 (1 self)
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We revisit a well studied linear algebraic problem, computing the rank and determinant of matrices, in order to obtain completeness results for small complexity classes. In particular, we prove that computing the rank of a class of diagonally dominant matrices is complete for L. We show that computing the permanent and determinant of tridiagonal matrices over Z is in GapNC 1 and is hard for NC 1. We also initiate the study of computing the rigidity of a matrix: the number of entries that needs to be changed in order to bring the rank of a matrix below a given value. We show that some restricted versions of the problem characterize small complexity classes. We also look at a variant of rigidity where there is a bound on the amount of change allowed. Using ideas from the linear interval equations literature, we show that this problem is NPhard over Q and that a certain restricted version is NPcomplete. Restricting the problem further, we obtain variations which can be computed in PL and are hard for C=L. 1
Forty necessary and sufficient conditions for regularity of interval matrices: A survey
 Electronic Journal of Linear Algebra
"... Abstract. This is a survey of forty necessary and sufficient conditions for regularity of interval matrices published in various papers over the last thirtyfive years. Afull list of references to the sources of all the conditions is given, and they are commented on in detail. ..."
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Abstract. This is a survey of forty necessary and sufficient conditions for regularity of interval matrices published in various papers over the last thirtyfive years. Afull list of references to the sources of all the conditions is given, and they are commented on in detail.
Parametric Interval Linear Solver
 Numerical Algorithms
"... Abstract. IntervalComputations‘LinearSystems ‘ is a Mathematica package supporting tools for solving parametric and nonparametric linear systems involving uncertainties. It includes a variety of functions, implementing different interval techniques, that help in producing sharp and rigorous results ..."
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Abstract. IntervalComputations‘LinearSystems ‘ is a Mathematica package supporting tools for solving parametric and nonparametric linear systems involving uncertainties. It includes a variety of functions, implementing different interval techniques, that help in producing sharp and rigorous results in validated interval arithmetic. The package is designed to be easy to use, versatile, to provide a necessary background for further exploration, comparisons and prototyping, and to provide some indispensable tools for solving parametric interval linear systems. This paper presents the functionality, provided by the current version of the package, and briefly discusses the underlying methodology. A new hybrid approach for sharp parametric enclosures, that combines parametric residual iteration, exact bounds, based on monotonicity properties, and refinement by interval subdivision, is outlined.
CONVEX SETS OF NONSINGULAR AND P–MATRICES (Linear and Multilinear Algebra, 38(3) : 233239, 1995)
, 2000
"... We show that the set r(A, B) (resp. c(A, B)) of square matrices whose rows (resp. columns) are independent convex combinations of ..."
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Cited by 2 (1 self)
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We show that the set r(A, B) (resp. c(A, B)) of square matrices whose rows (resp. columns) are independent convex combinations of
Generating and detecting matrices with positive principal minors
 Asian InformationScienceLife: An International Journal
, 2002
"... Abstract. A brief but concise review of methods to generate Pmatrices (i.e., matrices having positive principal minors) is provided and motivated by open problems on Pmatrices and the desire to develop and test efficient methods for the detection of Pmatrices. Also discussed are operations that l ..."
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Abstract. A brief but concise review of methods to generate Pmatrices (i.e., matrices having positive principal minors) is provided and motivated by open problems on Pmatrices and the desire to develop and test efficient methods for the detection of Pmatrices. Also discussed are operations that leave the class of Pmatrices invariant. Some new results and extensions of results regarding Pmatrices are included. Key words. Pmatrix, positive definite matrix, Mmatrix, MMAmatrix, Hmatrix, Bmatrix, totally positive matrix, mime, Schur complement, principal pivot transform, Cayley transform, linear complementarity problem, Pproblem. AMS subject classifications. 15A48, 15A15, 15A57, 1502, 9008 1. Introduction. An n × n complex matrix A ∈Mn(C) is called a Pmatrix if all its principal minors are positive. Recall that a principal minor is simply the determinant of a submatrix obtained from A when the same set of rows and columns are stricken out. The diagonal entries and the determinant of A are thus among its principal minors. We shall denote the class of complex Pmatrices by P.
PERRONFROBENIUS THEORY FOR COMPLEX MATRICES
"... Abstract. The purpose of this paper is to present a unified PerronFrobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The signreal spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with ..."
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Abstract. The purpose of this paper is to present a unified PerronFrobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The signreal spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with the Perron root of nonnegative matrices. In this paper we introduce the signcomplex spectral radius. Again, this quantity extends many properties of the Perron root of nonnegative matrices to general complex matrices. Various characterizations will be given, and many open problems remain. (1) 1. Introduction. The key to the generalizations of PerronFrobenius Theory to general real and to complex matrices is the following nonlinear eigenvalue problem: max{λ  : Ax  = λx, x � = 0}. Throughout the paper we use the notation that absolute value and comparison of vectors and matrices is always to be understood componentwise. For example, for C ∈ Mn(C) and
Improving interval enclosures
, 2009
"... This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval encl ..."
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This paper serves as background information for the Vienna proposal for interval standardization, explaining what is needed in practice to make competent use of the interval arithmetic provided by an implementation of the standard to be. Discussed are methods to improve the quality of interval enclosures of the range of a function over a box, considerations of possible hardware support facilitating the implementation of such methods, and the results of a simple interval challenge that I had posed to the reliable computing mailing list on November 26, 2008. Also given is an example of a bound constrained global optimization problem in 4 variables that has a 2dimensional continuum of global minimizers. This makes standard branch and bound codes extremely slow, and therefore may serve as a useful degenerate test problem.