The computation of Jordan blocks in parameter-dependent matrices

Akinola, Richard Olatokunbo; Freitag, Melina A.; Spence, A.

doi:10.1093/imanum/drt028

Cited by 7 publications

(10 citation statements)

References 22 publications

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“…One extremely important property is that information about derivatives of det(H(γ, ω)) = 0 is inherited by derivatives of f (γ, ω) = 0, and through this relationship information about the Jordan structure in H(γ, ω) may be extracted from derivatives of f (γ, ω) = 0. This is explained in detail in [2]. We continue this theme by now showing how Assumption 2.2 and Theorems 2.1 and 2.4 impact on a solution curve f (γ, ω) = 0.…”

Section: The Implicit Determinant Methodmentioning

confidence: 78%

See 1 more Smart Citation

Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

Freitag¹,

Spence²,

Dooren

2014

SIAM J. Matrix Anal. & Appl.

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We propose a fast algorithm to calculate the H∞-norm of a transfer matrix. The method builds on a well-known relationship between singular values of the transfer function and pure imaginary eigenvalues of a certain Hamiltonian matrix. Using this property we construct a two-parameter eigenvalue problem, where, in the generic case, the critical value corresponds to a two-dimensional Jordan block. We use the implicit determinant method which replaces the need for eigensolves by the solution of linear systems, a technique recently used in [M. A. Freitag and A. Spence, Linear Algebra Appl., 435 (2011), pp. 3189-3205] for finding the distance to instability. In this paper the method takes advantage of the structured linear systems that arise within the algorithm to obtain efficient solves using the staircase reduction. We give numerical examples and compare our method to the algorithm proposed in [N. Guglielmi, M. Gürbüzbalaban, and M. L.

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Section: The Implicit Determinant Methodmentioning

confidence: 78%

“…In this paper we introduce a new fast method for calculating this quantity by extending an algorithm recently introduced in [13] (see also [2,1,28]). …”

Section: Introductionmentioning

confidence: 99%

Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

Freitag¹,

Spence²,

Dooren

2014

SIAM J. Matrix Anal. & Appl.

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“…Indeed, an Hermitian problem with two varying parameters can admit singularity referred to as Diabolic point [45] which corresponds to a conical intersection of the eigenvalue loci surface. For real non-symmetric problems, EP can be also found and are strongly related to stability issues [28,29].…”

Section: Puiseux Seriesmentioning

confidence: 97%

“…It aims to identify the validity domain of such approaches and to extend it through analytic continuation. Eigenvalue perturbation is a widely spread mathematical problem [20,26] arising in many fields of application, ranging from noise attenuation in waveguides [27] to fluid-structure interaction like flutter [28,29]. These problems are inherently parametric and the behavior of eigenvalue loci, when parameters varies, has been widely studied for these applications in a deterministic framework.…”

Section: Introductionmentioning

confidence: 99%

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

Ghienne

Nennig

2020

Journal of Sound and Vibration

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A numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods. The radius of convergence of eigenvalue perturbation methods, based on Taylor series, is known to decrease when eigenvalues are getting closer to each other. This phenomenon, knwon as veering in structural dynamics, is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective. When this degeneracy, referred to as Exceptional Point (EP), is close to the real axis, the veering becomes stronger.The main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity. To do so, two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter. This yields a new robust eigenvalue reconstruction scheme which is compared to Taylor and Puiseux series. In all cases, theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model. This system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter. Computationally efficient, the proposed algorithm could also be relevant for actual numerical models of large size, arising from other applications involving parametric eigenvalue problems, e.g., waveguides, rotating machinery or instability problems such as squeal or flutter.

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“…p a = p p , and of the normal velocity, i.e. 1 iωρ a ∇p a · n = 1 iωρ p ∇p p · n, (32) must be satisfied. On rigid walls Γ w and Γ inc , the normal velocity vanishes and ∇p α · n = 0.…”

Section: Resonant Inclusion Embedded In a Porous Lined Duct 421 Fimentioning

confidence: 99%

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

Nennig

Perrey‐Debain

2020

Journal of Computational Physics

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A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex plane. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems.The method requires the computation of successive derivatives of two selected eigenvalues with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the localization of exceptional points (EP), using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order. This representation, which is associated with the topological structure of Riemann surfaces allows to efficiently approximate the selected pair in a certain neighbourhood of the EP.Practical applications dealing with guided acoustic waves propagating in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. The fact that EPs are associated with optimal dissipative treatments in the sense that they should provide best modal attenuation is also discussed.

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The computation of Jordan blocks in parameter-dependent matrices

Cited by 7 publications

References 22 publications

Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

Calculating the $H_{\infty}$-norm Using the Implicit Determinant Method

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

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