Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Corless, Robert M.; Rezvani, Nargol; Amiraslani, Amirhossein

doi:10.1007/s11786-007-0010-x

Cited by 11 publications

(6 citation statements)

References 16 publications

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“…To facilitate the investigation of the pseudospectra of matrix polynomials, [18] introduces and exploits a certain Möbius transformation named reversal with respect to a Lagrange basis.…”

Section: Definition 34 (Möbius Transformation)mentioning

confidence: 99%

“…A variety of transformations exploited in the literature [2,16,18,29,35,44,45,46,47,51] will be seen to be special instances of Möbius transformations. The broader theory we present here generalizes and unifies results that were hitherto observed for particular transformations, and provides a more versatile tool for investigating fundamental aspects of matrix polynomials.…”

Section: Introductionmentioning

confidence: 99%

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Möbius transformations of matrix polynomials

Mackey

Mehl

et al. 2015

Linear Algebra and its Applications

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We discuss Möbius transformations for general matrix polynomials over arbitrary fields, analyzing their influence on regularity, rank, determinant, constructs such as compound matrices, and on structural features including sparsity and symmetry. Results on the preservation of spectral information contained in elementary divisors, partial multiplicity sequences, invariant pairs, and minimal indices are presented. The effect on canonical forms such as Smith forms and local Smith forms, on relationships of strict equivalence and spectral equivalence, and on the property of being a linearization or quadratification are investigated. We show that many important transformations are special instances of Möbius transformations, and analyze a Möbius connection between alternating and palindromic matrix polynomials. Finally, the use of Möbius transformations in solving polynomial inverse eigenproblems is illustrated.

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“…To facilitate the investigation of the pseudospectra of matrix polynomials, [18] introduces and exploits a certain Möbius transformation named reversal with respect to a Lagrange basis.…”

Section: Definition 34 (Möbius Transformation)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Möbius transformations of matrix polynomials

Mackey

Mehl

et al. 2015

Linear Algebra and its Applications

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“…It has also been noted by both Tisseur and Higham [39] and Corless et al [11] that the norms A j 2 and P j 2 occurring in (3.4) and (3.5), respectively, can be replaced by nonnegative weights α j , not all equal to zero, which control how the perturbations to the coefficients are measured. Farouki and Rajan [17] have also defined these quantities as (absolute) condition numbers for the evaluation of polynomials (see also Corless and Fillion [10, Thm.…”

Section: Linearization One Of the Most Widespread Solution Methods Fmentioning

confidence: 97%

“…Amiraslani [1] and Corless et al [11] have also extended this result to the Lagrange basis by considering the ε-pseudospectrum of polynomials expressed in other bases. The equivalent expressions (for the 2-norm) for the backward errors are obtained by replacing…”

Section: Linearization One Of the Most Widespread Solution Methods Fmentioning

confidence: 99%

Backward Error of Polynomial Eigenvalue Problems Solved by Linearization of Lagrange Interpolants

Lawrence

Corless²

2015

SIAM J. Matrix Anal. & Appl.

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This article considers the backward error of the solution of polynomial eigenvalue problems expressed as Lagrange interpolants. One of the most common strategies to solve polynomial eigenvalue problems is to linearize, which is to say that the polynomial eigenvalue problem is transformed into an equivalent larger linear eigenvalue problem, and solved using any appropriate eigensolver. Much of the existing literature on the backward error of polynomial eigenvalue problems focuses on polynomials expressed in the classical monomial basis. Hence, the objective of this article is to carry out the necessary extensions for polynomials expressed in the Lagrange basis. We construct one-sided factorizations that give simple expressions relating the eigenvectors of the linearization to the eigenvectors of the polynomial eigenvalue problem. Using these relations, we are able to bound the backward error of an approximate eigenpair of the polynomial eigenvalue problem relative to the backward error of an approximate eigenpair of the linearization. We develop bounds for the backward error involving both the norms of the polynomial coefficients and the properties of the Lagrange basis generated by the interpolation nodes. We also present numerous numerical examples to illustrate the numerical properties of the linearization, and develop a balancing strategy to improve the accuracy of the computed solutions. Abstract. This article considers the backward error of the solution of polynomial eigenvalue problems expressed as Lagrange interpolants. One of the most common strategies to solve polynomial eigenvalue problems is to linearize, which is to say that the polynomial eigenvalue problem is transformed into an equivalent larger linear eigenvalue problem, and solved using any appropriate eigensolver. Much of the existing literature on the backward error of polynomial eigenvalue problems focuses on polynomials expressed in the classical monomial basis. Hence, the objective of this article is to carry out the necessary extensions for polynomials expressed in the Lagrange basis. We construct one-sided factorizations that give simple expressions relating the eigenvectors of the linearization to the eigenvectors of the polynomial eigenvalue problem. Using these relations, we are able to bound the backward error of an approximate eigenpair of the polynomial eigenvalue problem relative to the backward error of an approximate eigenpair of the linearization. We develop bounds for the backward error involving both the norms of the polynomial coefficients and the properties of the Lagrange basis generated by the interpolation nodes. We also present numerous numerical examples to illustrate the numerical properties of the linearization, and develop a balancing strategy to improve the accuracy of the computed solutions.

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“…In this section we review some of the main properties of generalized eigenvalues to be used in the next sections. See Amiraslani et al (2009); Corless et al (2007) for a more detailed discussion.…”

Section: Generalized Eigenvalue Problemsmentioning

confidence: 99%

Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”

Corless

Diaz–Toca

Fioravanti

et al. 2013

Computer Aided Geometric Design

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This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only "by values". This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points.

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Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Cited by 11 publications

References 16 publications

Möbius transformations of matrix polynomials

Möbius transformations of matrix polynomials

Backward Error of Polynomial Eigenvalue Problems Solved by Linearization of Lagrange Interpolants

Computing the topology of a real algebraic plane curve whose defining equations are available only “by values”

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