An improved grid method for the computation of the pseudospectra of matrix polynomials

Fatouros, Stavros; Psarrakos, Panayiotis

doi:10.1016/j.mcm.2008.05.047

Cited by 3 publications

(2 citation statements)

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“…Note this yields a sufficiently accurate sketch of σ ,w (P) and is very competitive when compared to other methods. For instance, Figure 5b is obtained via the procedure in [31] applied to a 400 × 400 grid on the relevant region 15,15]. This latter approach is far more computationally intensive, requiring 71,575 iterations to visualize σ ,w (P) for the same tuple of parameters.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Eigenvalue Estimates via Pseudospectra

2021

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In this note, given a matrix A∈Cn×n (or a general matrix polynomial P(z), z∈C) and an arbitrary scalar λ0∈C, we show how to define a sequence μkk∈N which converges to some element of its spectrum. The scalar λ0 serves as initial term (μ0=λ0), while additional terms are constructed through a recursive procedure, exploiting the fact that each term μk of this sequence is in fact a point lying on the boundary curve of some pseudospectral set of A (or P(z)). Then, the next term in the sequence is detected in the direction which is normal to this curve at the point μk. Repeating the construction for additional initial points, it is possible to approximate peripheral eigenvalues, localize the spectrum and even obtain spectral enclosures. Hence, as a by-product of our method, a computationally cheap procedure for approximate pseudospectra computations emerges. An advantage of the proposed approach is that it does not make any assumptions on the location of the spectrum. The fact that all computations are performed on some dynamically chosen locations on the complex plane which converge to the eigenvalues, rather than on a large number of predefined points on a rigid grid, can be used to accelerate conventional grid algorithms. Parallel implementation of the method or use in conjunction with randomization techniques can lead to further computational savings when applied to large-scale matrices.

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Section: Numerical Experimentsmentioning

confidence: 99%

Eigenvalue Estimates via Pseudospectra

2021

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“…This approach is now linked to the notion of pseudo-spectrum developed in (Tisseur & Higham, 2001), ( Lancaster & Psarrakos, 2005 The (weighted) ε-pseudo-spectrum of ( ) P s (Tisseur & Higham, 2001) is then defined as open disk that does not intersect the boundaries of the pseudo-spectrum (Psarrakos, 2007), (Fatouros & Psarrakos, 2009). Based on that an algorithm that uses exclusion disks is introduced in (Fatouros & Psarrakos, 2009) for the computation of the pseudo-spectra. …”

Section: Formentioning

confidence: 99%

Approximate Greatest Common Divisor of Many Polynomials and Pseudo-Spectrum

Fatouros

Karcanias

Christou

et al. 2013

IFAC Proceedings Volumes

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This is the accepted version of the paper.This version of the publication may differ from the final published version. (e-mail: N.Karcanias@city.ac.). Permanent Abstract:The paper is concerned with establishing the links between the approximate GCD of a set of polynomials and the notion of the pseudo-spectrum defined on a set of polynomials. By examining the pseudo-spectrum of the structured matrix we will derive estimates of the area of the approximate roots of the initial polynomial set. We will relate the strength of the GCD to the weighted strength of the pseudospectra and we investigate under which conditions the roots of the approximate GCDs are a subset of the pseudo-spectra.

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