On the interval eigenvalue problem

Milan Hladík

Abstract

(joint work with D. Daney and E. P. Tsigaridas)

An interval matrix is defined as a matrix whose entries vary inside given intervals. Naturally, its real eigenvalues also range in some intervals. The interval eigenvalue problem is to determine exactly or give an outer approximation of the eigenvalue set. This problem has many application in the field of mechanics and engineering (among others), and is considered to be hard. Indeed, only checking if zero is contained in the eigenvalue set is NP-hard.

We present recent results in this subject. First, we review and improve some formulae for initial approximation of the eigenvalue set; they are usually quickly computable at the expense of tightness. Then a new filtering method for making an initial approximation tighter is proposed. Next we present a branch & prune algorithm that approximates the eigenvalue set with a given accuracy. Moreover, exact bounds (limited by the use of