Backward error and condition of polynomial eigenvalue problems

Tisseur, Françoise

doi:10.1016/s0024-3795(99)00063-4

Cited by 219 publications

(246 citation statements)

References 22 publications

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“…Note also that the backward error analysis of standard linearization processes indicates that to compute eigenvalues with small backward errors, it is important for the nonzero blocks of the linearization to have norm close to one (see [12,14,21]). We show next that a possible choice for the points σ i , i = 1, .…”

Section: Choice Of the Pointsmentioning

confidence: 99%

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Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Barel

Tisseur

2018

Linear Algebra and its Applications

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We propose a novel approach to solve polynomial eigenvalue problems via linearization. The novelty lies in (a) our choice of linearization, which is constructed using input from tropical algebra and the notion of well-separated tropical roots, (b) an appropriate scaling applied to the linearization and (c) a modified stopping criterion for the QZ iterations that takes advantage of the properties of our scaled linearization. Numerical experiments show that our polynomial eigensolver computes all the finite and well-conditioned eigenvalues to high relative accuracy even when they are very different in magnitude.

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Section: Choice Of the Pointsmentioning

confidence: 99%

“…Following [21] the backward error η P ( λ) for a computed finite eigenvalue λ of P can be computed as…”

Section: Numerical Experimentsmentioning

confidence: 99%

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Barel

Tisseur

2018

Linear Algebra and its Applications

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“…Unless the block structure of the linearization is respected (and it is not by standard algorithms), the conditioning of the larger linear problem can be worse than that of the original matrix polynomial, since the class of admissible perturbations is larger. For example, eigenvalues that are wellconditioned for P(λ ) may be ill-conditioned for L(λ ) [39,41,78]. Ideally, when solving (20) via (21) we would like to have κ P (λ ) ≈ κ L (λ ).…”

Section: Impact On Numerical Practicementioning

confidence: 99%

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Mackey

Tisseur

2015

Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory

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Matrix polynomial eigenproblems arise in many application areas, both directly and as approximations for more general nonlinear eigenproblems. One of the most common strategies for solving a polynomial eigenproblem is via a linearization, which replaces the matrix polynomial by a matrix pencil with the same spectrum, and then computes with the pencil. Many matrix polynomials arising from applications have additional algebraic structure, leading to symmetries in the spectrum that are important for any computational method to respect. Thus it is useful to employ a structured linearization for a matrix polynomial with structure. This essay surveys the progress over the last decade in our understanding of linearizations and their construction, both with and without structure, and the impact this has had on numerical practice.

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“…The backward error and conditioning of polynomial eigenvalue problems expressed in the monomial basis is studied in [20]. Backward error and conditioning for scalar polynomials expressed in other bases are studied in many places, for example in [8].…”

Section: Pseudospectra and Conditioningmentioning

confidence: 99%

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

2007

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Abstract. Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of "good" nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied.

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Backward error and condition of polynomial eigenvalue problems

Cited by 219 publications

References 22 publications

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Polynomial eigenvalue solver based on tropically scaled Lagrange linearization

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

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