Nonlinear Eigenvalue Problems with Specified Eigenvalues

Karow, Michael; Kreßner, Daniel; Mengi, Emre

doi:10.1137/130927462

Cited by 5 publications

(1 citation statement)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A lower bound is found for the general case when all coefficient matrices are perturbed. The techniques are further extended to find the same distance for more general nonlinear eigenvalue problems in [6].…”

Section: Introductionmentioning

confidence: 99%

Nearest matrix polynomials with a specified elementary divisor

Das¹,

Bora²

2019

Preprint

View full text Add to dashboard Cite

The problem of finding the distance from a given n × n matrix polynomial of degree k to the set of matrix polynomials having the elementary divisor (λ−λ0) j , j r, for a fixed scalar λ0 and 2 r kn is considered. It is established that polynomials that are not regular are arbitrarily close to a regular matrix polynomial with the desired elementary divisor. For regular matrix polynomials the problem is shown to be equivalent to finding minimal structure preserving perturbations such that a certain block Toeplitz matrix becomes suitably rank deficient. This is then used to characterize the distance via two different optimizations. The first one shows that if λ0 is not already an eigenvalue of the matrix polynomial, then the problem is equivalent to computing a generalized notion of a structured singular value. The distance is computed via algorithms like BFGS and Matlab's globalsearch algorithm from the second optimization. Upper and lower bounds of the distance are also derived and numerical experiments are performed to compare them with the computed values of the distance.

show abstract

Section: Introductionmentioning

confidence: 99%