Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Abstract

Abstract. We study bounds on real eigenvalues of interval matrices, and our aim is to develop fast computable formulae that produce as-sharp-as-possible bounds. We consider two cases: general and symmetric interval matrices. We focus on the latter case, since on the one hand such interval matrices have many applications in mechanics and engineering, and on the other hand many results from classical matrix analysis could be applied to them. We also provide bounds for the singular values of (generally nonsquare) interval matrices. Finally, we illustrate and compare the various approaches by a series of examples.

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

“…A number of methods have been proposed in the literature to obtain lower and upper bounds on the smallest and largest eigenvalues, respectively, of interval matrices [1,7,9,[23][24][25]. Other methods have been devised to compute bounds for each individual eigenvalue [9,11,12,21].…”

“…Other methods have been devised to compute bounds for each individual eigenvalue [9,11,12,21]. An evolutionary method approach for inner bounds was presented by Yuan et al [27].…”

“…The branching occurs on the interval entries of the input matrix. We use Ronh's theorem [9,24], which is an interval extension of Weyl's theorem [6] and local improvement methods in order to obtain valid bounds at each step. The algorithm can be used to calculate the bounds of a specific eigenvalue regardless of whether its range overlaps with that of other eigenvalues or not.…”

“…(5) while iter ≤ maxiters do (6) Choose the first entry, L 1 , from list L. [2], and u = L 1 [3]. (8) Delete L 1 from L. (9) if l < BUB then (10) Choose branching entry [m ij ], i = j. (12) Obtain bounds l 1 and u 1 for λ k ([M 1 ]).…”

“…We can write any given interval matrix [M] Theorem 4.1 (Rohn [9,24]) Given a symmetric interval matrix…”