Differential Equations for Real-Structured Defectivity Measures

Buttà, Paolo; Guglielmi, Nicola; Manetta, Manuela; Noschese, Silvia

doi:10.1137/140964631

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…When

scriptS = C^{n \times n}

, i.e., when A has no particular structure, the perturbations that affect the eigenvalue λ of A the most, relative to the norm of the perturbation, are multiples of the rank‐one matrices . The Wilkinson disks for the different eigenvalues are disjoint if the radius t of the disks is smaller than the distance ϵ ∗ from defectivity of the matrix A ,

ϵ_{*} = \inf {∥ A - B ∥_{F} : B \in {double-struckC}^{n \times n} is defective} .

Analogously, in case

scriptS \underset{\neq}{\subset} C^{n \times n}

the threshold is the structured distance from defectivity

ϵ_{*}^{scriptS}

of A ,

ϵ_{*}^{S} = \inf {∥ A - B ∥_{F} : B \in S is defective} .

Clearly,

ϵ_{*}^{scriptS} ⩾ ϵ_{*}

;see, e.g., for details. In the structured case, the rank‐one Wilkinson perturbations are projected as described in Section 3.…”

Section: Approximated Structured ϵ‐Pseudospectramentioning

confidence: 98%

Approximated structured pseudospectra

Noschese

Reichel

2016

Numerical Linear Algebra App

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Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small-matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra

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“…When

scriptS = C^{n \times n}

ϵ_{*} = \inf {∥ A - B ∥_{F} : B \in {double-struckC}^{n \times n} is defective} .

Analogously, in case

scriptS \underset{\neq}{\subset} C^{n \times n}

the threshold is the structured distance from defectivity

ϵ_{*}^{scriptS}

of A ,

ϵ_{*}^{S} = \inf {∥ A - B ∥_{F} : B \in S is defective} .

Clearly,

ϵ_{*}^{scriptS} ⩾ ϵ_{*}

;see, e.g., for details. In the structured case, the rank‐one Wilkinson perturbations are projected as described in Section 3.…”

Section: Approximated Structured ϵ‐Pseudospectramentioning

confidence: 98%

Approximated structured pseudospectra

Noschese

Reichel

2016

Numerical Linear Algebra App

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show abstract

“…, [1,2,9,10] for details. In the structured case, the rank-one Wilkinson perturbations (2.1) are projected as described in Section 3.…”

Section: Approximated Structured ε-Pseudospectramentioning

confidence: 99%

Approximated structured pseudospectra

Noschese¹,

Reichel²

2016

Preprint

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Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra.

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“…This completes the proof. Similarly to [BGMN13] we state the following uniqueness result for sufficiently small ε.…”

mentioning

confidence: 93%

Computing Extremal Points of Symplectic Pseudospectra and Solving Symplectic Matrix Nearness Problems

Guglielmi¹,

Kreßner²,

Lubich³

2014

SIAM J. Matrix Anal. & Appl.

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We study differential equations that lead to extremal points in symplectic pseudospectra. In a two-level approach, where on the inner level we compute extremizers of the symplectic ε-pseudospectrum for a given ε and on the outer level we optimize over ε, this is used to solve symplectic matrix nearness problems such as the following: For a symplectic matrix with eigenvalues of unit modulus, we aim to determine the nearest complex symplectic matrix such that some or all eigenvalues leave the complex unit circle. Conversely, for a symplectic matrix with all eigenvalues lying off the unit circle, we consider the problem of computing the nearest symplectic matrix that has an eigenvalue on the unit circle.

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Differential Equations for Real-Structured Defectivity Measures

Cited by 5 publications

References 17 publications

Approximated structured pseudospectra

Approximated structured pseudospectra

Approximated structured pseudospectra

Computing Extremal Points of Symplectic Pseudospectra and Solving Symplectic Matrix Nearness Problems

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