Jiri Rohn

Systems of linear interval equations

This paper presents theory and methods for computing the exact bounds on the solution of a system of n linear equations in n variables whose coefficients and right-hand sides vary in some real intervals. Finite and iterative methods are given, based on results from linear complementarity theory. Also regularity conditions, regularity testing, and computing the exact inverse of an interval matrix are dealt with.

Positive Definiteness and Stability of Interval Matrices

Characterizations of positive definiteness, positive semidefiniteness, and Hurwitz and Schur stability of interval matrices are given. First it is shown that an interval matrix has some of the four properties if and only if this is true for a finite subset of explicitly described matrices, and some previous results of this type are improved. Second it is proved that a symmetric interval matrix is positive definite (Hurwitz stable, Schur stable) if and only if it contains at least one symmetric matrix with the respective property and is nonsingular (for Schur stability, two interval matrices are to be nonsingular). As a consequence, verifiable sufficient conditions are obtained for positive definiteness and Hurwitz and Schur stability of symmetric interval matrices.

Computing Exact Componentwise Bounds on Solutions of Linear Systems with Interval Data isNP-Hard

We prove that it is NP-hard to compute the exact componentwise bounds on solutions of all the linear systems that can be obtained from a given linear system with a nonsingular matrix by perturbing all the data independently of each other within prescribed tolerances.

A theorem of the alternatives for the equation Ax + B|x| = b

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B 0|x| = 0 has a nontrivial solution for some matrix B 0 of a very special form, |B 0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B 0|x| = 0 has a nontrivial solution for some matrix B 0 of a very special form, |B 0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.

Sufficient Conditions for Regularity and Singularity of Interval Matrices

Several verifiable sufficient conditions for regularity and singularity of interval matrices are given. As an application, a verifiable sufficient condition is derived for an interval matrix to have all eigenvalues real.

Inverse interval matrix

The inverse interval matrix is defined as the narrowest interval matrix containing the inverses of all the matrices from a given interval matrix. Both theoretical and practical results concerning computation of the inverse interval matrix are presented. In particular, explicit formulas are given for the inverse of an interval matrix with radius of rank one.

AN ALGORITHM FOR SOLVING THE ABSOLUTE VALUE EQUATION

Presented is an algorithm which for each A,B ∈ Rn×n and b ∈ Rn in a finite number of steps either finds a solution of the equation Ax+B|x| = b, or states existence of a singular matrix S satisfying |S − A |≤| B| (and in most cases also constructs such an S).

Interval matrices: singularity and real eigenvalues

This paper proves that a singular interval matrix contains a singular matrix of a very special form. This result is applied to study the real part L of the spectrum of an interval matrix. Under the assumption of sign stability of eigenvectors this paper gives a complete description of L by means of spectra of a finite subset of matrices and formulates a stability criterion for interval matrices with real eigenvalues that requires checking only two matrices for stability.

New condition numbers for matrices and linear systems

New condition numbers for matrices and linear systems are proposed, based on the dependence of the relative errors in the result upon the relative errors of the data.ZusammenfassungNeue Konditionszahlen für Matrizen und lineare Gleichungssysteme werden eingeführt, die auf der Abhängigkeit der relativen Fehler des Resultats von den relativen Datenfehlern basieren.

An Algorithm for Checking Regularity of Interval Matrices

Checking regularity (or singularity) of interval matrices is a known NP-hard problem. In this paper a general algorithm for checking regularity/singularity is presented which is not a priori exponential. The algorithm is based on a theoretical result according to which regularity may be judged from any single component of the solution set of an associated system of linear interval equations. Numerical experiments (with interval matrices up to the size n = 50) confirm that this approach brings an essential decrease in the amount of operations needed.