An evolution strategy method for computing eigenvalue bounds of interval matrices

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.

show abstract

“…Theoretical results could also be found in the work of Qiu and Wang [28]. However, some results turned out to be incorrect [34].…”

mentioning

confidence: 52%

“…An evolution strategy method by Yuan, He, and Leng [34] yields an inner approximation of the eigenvalue Downloaded 01/10/13 to 138.96.201.247. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php set.…”

mentioning

confidence: 99%

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Hladík¹,

Daney²,

Tsigaridas³

2010

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…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]. 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.…”

Section: Nerantzis and Cs Adjimanmentioning

confidence: 99%

“…(24) end if (25) [2] being the second entry of the first sublist in L. (26) iter++. (27) if L is empty or BUB − BLB < then (28) Return BLB and BUB. (29) end if (30) end if (31) end while (32) Return BLB and BUB.…”

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

View full text Add to dashboard Cite

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.

show abstract

“…Due to intractability of the above formulae for larger dimensions, there were developed various approximation methods. Enclosure methods for the eigenvalue set, which yield lower bounds on min comprise, e.g., [7,16,17]. The related topic of finding verified intervals of eigenvalues for real matrices was studied in e.g.…”

Section: Introductionmentioning

confidence: 99%

Complexity issues for the symmetric interval eigenvalue problem

Hladík

2014

Open Mathematics

View full text Add to dashboard Cite

Abstract:We study the problem of computing the maximal and minimal possible eigenvalues of a symmetric matrix when the matrix entries vary within compact intervals. In particular, we focus on computational complexity of determining these extremal eigenvalues with some approximation error. Besides the classical absolute and relative approximation errors, which turn out not to be suitable for this problem, we adapt a less known one related to the relative error, and also propose a novel approximation error. We show in which error factors the problem is polynomially solvable and in which factors it becomes NP-hard.

show abstract

An evolution strategy method for computing eigenvalue bounds of interval matrices

Cited by 22 publications

References 13 publications

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

Bounds on Real Eigenvalues and Singular Values of Interval Matrices

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

Complexity issues for the symmetric interval eigenvalue problem

Contact Info

Product

Resources

About