Computing bounds to real eigenvalues of real‐interval matrices

Leng, Huinan; He, Zhiqing; Yuan, Quan

doi:10.1002/nme.2179

Cited by 28 publications

(9 citation statements)

References 19 publications

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“…The algorithms by Hladík et al [10], Leng et al [15], and Leng [14] can be used to calculate the real eigenvalue set of an interval matrix with any given precision. These algorithms begin with the calculation of an initial inclusion set and proceed by successive identification and removal of parts of the initial inclusion set which do not belong to the eigenvalue set.…”

Section: Nerantzis and Cs Adjimanmentioning

confidence: 99%

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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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Section: Nerantzis and Cs Adjimanmentioning

confidence: 99%

“…In particular, the algorithm by Hladík et al has been shown to be fast and applicable to very large matrices (with small interval widths). However, when the ranges of individual eigenvalues overlap, the methods in [10,14,15] can only provide, at best, the bounds of the union of the overlapping ranges.…”

Section: Nerantzis and Cs Adjimanmentioning

confidence: 99%

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

Nerantzis

Adjiman

2016

Optimization Methods and Software

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“…Calculate the eigenvalue bounds for the symmetric tridiagonal interval matrix [19] (a 11 , a 12 , a 22 , a 23 , a 33 , a 34 , a 44 , a 45 …”

Section: Examplementioning

confidence: 99%

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

Yuan

Leng

2010

Numer. Linear Algebra Appl.

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The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example.A k+1,nThe eigenvalues of W k are denoted and ordered as 1 2 ··· n−1 .

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“…Hence from Theorem 2.1, the bounds of its interval eigenvalues should be reached at some vertex matrices. Now the method in 19 is applied to solve the problem. The results are shown in Table I.…”

Section: Numerical Experimentsmentioning

confidence: 99%