Systems of linear interval equations

Certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined. We extend a proposed technique of Hansen and Rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix $ {\bf{A}}= [ \underline A, \overline A ]$ is an H-matrix; when ${\bf A}$ is centered about a diagonal matrix, these bounds are the smallest possible (i.e., the bounds are then the solution hull). Hansen's scheme also computes the solution hull when the linear interval system ${\bf{A}}{\bf{x}}={\bf{b}}=[{\underline b}, {\overline b}]$ is such that ${\bf{A}}$ is inverse positive and ${\underline b}=- {\overline b} \neq 0$. Earlier results of others also imply that, when ${\bf{A}}$ is an M-matrix and ${\bf{b}} \geq 0, {\bf{b}} \leq 0$, or $0 \in {\bf{b}}$, interval Gaussian elimination (IGA) computes the hull. We also give a method of computing the solution hull inexpensively in many instances when ${\bf A}$ is inverse positive, given an outer approximation such as that obtained from IGA. Examples are used to compare these schemes under various conditions.

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A Comparison of some Methods for Solving Linear Interval Equations

By using a new set of arithmetic operations on interval numbers, we discuss some arithmetic properties of interval matrices which intern helps us to compute the powers of interval matrices and to solve the system of interval linear equations. Keywords—Interval arithmetic, Interval matrix, linear equations.

On Some Properties of Interval Matrices

This is a survey of forty necessary and sufficient conditions for regularity of interval matrices published in various papers over the last thirty-five years. Afull list of references to the sources of all the conditions is given, and they are commented on in detail. 1. Introduction. During the last thirty-five years (1973-2008), considerable in- terest has been dedicated to the problemof regularity of interval m atrices. It has resulted in formulations of altogether forty necessary and sufficient conditions that constitute the subject matter of this survey paper. By definition, a square interval matrix A is called regular if each A ∈ A is nonsingular, and it is said to be singular otherwise (i.e., if it contains a singular matrix). It is the purpose of this paper to show that this property can be reformulated in surprisingly many surprisingly various ways. In the main Theorem 4.1 we show that regularity of interval matrices can be characterized in terms of determinants (Theorem 4.1, condition (xxxii)), matrix inverses (xxx), linear equations (xxv), absolute value equations (v), absolute value inequalities (ii), matrix equations (xxiv), solvability in each orthant (xvi), inclusions (xxxvii), set properties (xxxvi), real spectral radius (xxxiv), P -matrices (xxix) and edge nonsingularity (xli). We do not include the proof of mutual equivalence of all the conditions since that would make for a very lengthy

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Forty necessary and sufficient conditions for regularity of interval matrices: A survey

Shape recognition using eigenvalues of the Dirichlet Laplacian

The fixed membrane problem All + Xu = 0 in Q, u = 0 on aQ, for a bounded region Q of the plane, is approximated by a finite-difference scheme whose matrix is monotone. By an extension of previous methods for schemes with matrices of positive type, 0(h 4) convergence is shown for the approximating eigenvalues and eigenfunctions, where h is the mesh width. An application to an approximation of the forced vibration problem Au + qu = J in Q, u = 0 in aQ, is also given.

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J. R. Kuttler

Papers

A fourth-order finite-difference approximation for the fixed membrane eigenproblem