Characterizing and approximating eigenvalue sets of symmetric interval matrices

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guarantee exactness. We illustrate our approach by several examples and numerical experiments. Key-words: Interval matrix, symmetric matrix, interval analysis, eigenvalue, eigenvalue bounds ∗ Charles University, Faculty of Mathematics and Physics, Department of Applied Mathematics, Malostranské nám. 25, 11800, Prague, Czech Republic, e-mail: [email protected] † INRIA Sophia-Antipolis Méditerranée, 2004 route des Lucioles, BP 93, 06902 SophiaAntipolis Cedex, France, e-mail: FirstName.LastName(AT)inria.fr ‡ Computer Science Department, Aarhus University, Denmark. e-mail: [email protected] Caractérisation et approximation des ensembles de valeurs propres des matrices symétriques intervalles Résumé : We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guarantee exactness. We illustrate our approach by several examples and numerical experiments. Mots-clés : Matrice d’intervalles, matrice symétrique, analyse par intervalles, valeur propre, borne sur les valeurs propres Characterizing and approximating eigenvalue sets of symmetric interval matrices3