Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

Abstract

The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝn×n for the general case where the matrix elements aij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λμ, x(μ)), μ = 1, ..., n, is suggested for the case where λμ is a simple eigenvalue. It is based on the use of an affine interval approximation of a ij (p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.