An Overview of Polynomially Computable Characteristics of Special Interval Matrices

TL;DR

This paper reviews classes of special interval matrices with polynomially computable properties, focusing on their determinants, eigenvalues, and norms, and presents both known facts and new insights.

Contribution

It provides a comprehensive survey of polynomially solvable classes of special interval matrices, including new perspectives on their properties and computations.

Findings

Polynomially computable determinants and eigenvalues for certain matrix classes

Explicit formulas for inverse matrices of special interval matrices

Characterization of solution set hulls for interval linear systems

Abstract

It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying polynomially solvable classes thus belongs to important current trends. The purpose of this paper is to review some of such classes. In particular, we focus on several special interval matrices and investigate their convenient properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse M-matrices, inverse nonnegative matrices, nonnegative matrices, totally positive matrices and some others. We focus in particular on computing the range of the determinant, eigenvalues, singular values, and selected norms. Whenever possible, we state also formulae for determining the inverse matrix and the hull of the solution set of an interval system of linear equations. We survey not only the known facts, but we present some new views as well.