The Interval Eigenvalue Problem

Deif, A. S.

doi:10.1002/zamm.19910710117

Cited by 68 publications

(46 citation statements)

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“…The problem of computing lower and upper bounds for the eigenvalue set is well studied; see, e.g., [3,10,17,27,28,29,30,32]. In recent years some effort was made in developing and extending diverse inclusion sets for eigenvalues [8,22] such as Gerschgorin discs or Cassini ovals.…”

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confidence: 99%

“…For other approaches that estimate Λ, we refer the reader to [10,27,32]. Some methods do not calculate bounds for the real eigenvalues of A; instead they compute bounds for the real parts of the complex eigenvalues.…”

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confidence: 99%

“…Such a result could be of extreme practical importance since it can be used for testing the accuracy of existing approximation algorithms. In this line of research, let us mention the work of Deif [10] and Hertz [15,16]. The former provides an exact description of the eigenvalue set, but it works only under some not easily verified assumptions on sign pattern invariance of eigenvectors; the latter (see also [31]) proposes a formula for computing the exact extremal values λ 1 (A S ), λ 1 (A S ), λ n (A S ), and λ n (A S ), which consists of 2 n−1 iterations.…”

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confidence: 99%

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Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Hladík¹,

Daney²,

Tsigaridas³

2010

SIAM J. Matrix Anal. & Appl.

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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.

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Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Hladík¹,

Daney²,

Tsigaridas³

2010

SIAM J. Matrix Anal. & Appl.

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“…An evolutionary method approach for inner bounds was presented by Yuan et al [27]. Exact bounds for individual eigenvalues have been given by Deif [3] provided that the signs of the eigenvector entries remain constant over the interval matrix. This condition limits the applicability of this result.…”

Section: Nerantzis and Cs Adjimanmentioning

confidence: 99%

“…(4) Initialize L = {{[M], l, u}}, iter = 0. (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.…”

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

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

Nerantzis

Adjiman

2016

Optimization Methods and Software

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We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the kth largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn's method (an interval extension of Weyl's theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision > 0 (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated.

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