On the quadratic two-parameter eigenvalue problem and its linearization

Muhič, Andrej; Plestenjak, Bor

doi:10.1016/j.laa.2009.12.022

Cited by 27 publications

(47 citation statements)

References 9 publications

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“…In this case, as shown in [14], if all eigenvalues of (1) are algebraically simple, which is the generic case, they agree with the finite regular eigenvalues of the pair of singular matrix pencils (3). A numerical algorithm from [15], based on the staircase algorithm by Van Dooren [20], may then be used to compute the common regular part of (3) and extract the finite regular eigenvalues. The algorithm returns matrices Q and P with orthonormal columns such that the finite regular eigenvalues are the eigenvalues of the following pair of generalized eigenvalue problems…”

Section: Introductionmentioning

confidence: 89%

“…From Proposition 7 we know that the operator Q diag (P 1 (θ, η), P 2 (θ, η)) Q is asymptotically invertible. Moreover, from (15) it is clear that, neglecting higher-order terms, Therefore,…”

Section: Propositionmentioning

confidence: 99%

“…We will discuss both of these applications in Section 2. In the special case that k = 2 we speak of the quadratic multiparameter eigenvalue problem (QMEP); this problem has already been discussed in [6,11,15].…”

Section: Introductionmentioning

confidence: 99%

“…It is shown in [15] that, analogously to one-parameter matrix polynomials, a PMEP (1) can be linearized as a two-parameter eigenvalue problem (MEP) of the form (2) W 1 (λ, µ) z 1 = (A 1 + λB 1 + µC 1 ) z 1 = 0,…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Hochstenbach

Muhič

Plestenjak

2015

Journal of Computational and Applied Mathematics

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Section: Introductionmentioning

confidence: 89%

Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Hochstenbach

Muhič

Plestenjak

2015

Journal of Computational and Applied Mathematics

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“…Unless (d 1 , d 2 ) = (N 1 , N 2 ), both pencils in (12) are singular. In this case we first apply the staircase algorithm from [28] …”

Section: Numerical Experimentsmentioning

confidence: 99%

Uniform Determinantal Representations

Boralevi

Doornmalen²,

Draisma

et al. 2017

SIAM J. Appl. Algebra Geometry

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Abstract. The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.

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A design procedure for buildings equipped with energy dissipation devices using nonclassical damping and iso‐performance curves

Llera

Muñoz

Besa

2018

Earthq Engng Struct Dyn

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Summary This article describes a design procedure for elastic buildings equipped with linear and nonlinear energy dissipating devices. The objective is to achieve a design that responds to a target building performance following a simple and robust step‐by‐step algorithm. The proposed procedure identifies first the modal significance of key design performance indicators and controls the modal properties by solving a singular two‐parameter eigenvalue problem. For that purpose, a new modal significance metric is proposed, and a target frequency shift and damping ratio for the complete structure are obtained from the so‐called iso‐performance design curves. The design algorithm employs linear‐equivalent stiffness and damping properties, which are then transformed into parameters characterizing inelastic force‐deformation constitutive models corresponding to physical devices. The design algorithm leads to an optimal damper distribution corresponding to the minimum global amount of supplemental equivalent damping needed to achieve a maximum modal perturbation. The design procedure is first demonstrated using a five‐story building example and then a real and complex 22‐story free‐plan building with two towers of rhomboid‐shape plan with a very singular dynamic behavior.

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On the quadratic two-parameter eigenvalue problem and its linearization

Cited by 27 publications

References 9 publications

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems

Uniform Determinantal Representations

A design procedure for buildings equipped with energy dissipation devices using nonclassical damping and iso‐performance curves

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