A Comparison of some Methods for Solving Linear Interval Equations

Abstract

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.