Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures

Bora, Shreemayee; Karow, Michael; Mehl, Christian; Sharma, Punit

doi:10.1137/130925621

Cited by 19 publications

(28 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This has lead to the development of a number of structure-preserving algorithms for structured linearizations of structured eigenproblems [45,47,65,68,70,75,79], as well as the derivation of structured backward errors and structured condition numbers corresponding to structured pertubations [1,2,3,11,12].…”

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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…We consider perturbations that leave certain coefficient matrices of P (z) unchanged to be elements of some subset of (C n×n ) p , where p < m is a positive integer determined by the number of coefficient matrices of P (z) that are perturbed. We follow the strategy used in [6] to reformulate the problem of computing η S w (P, λ) in terms of a structured mapping problem. A key result in this respect is [6,Lemma 2.4].…”

Section: Preliminariesmentioning

confidence: 99%

“…We follow the strategy used in [6] to reformulate the problem of computing η S w (P, λ) in terms of a structured mapping problem. A key result in this respect is [6,Lemma 2.4]. It states that if any λ ∈ C is not an eigenvalue of an n × n matrix polynomial P (z) = m j=0 z j A j , then λ is an eigenvalue of a perturbed polynomial and…”

Section: Preliminariesmentioning

confidence: 99%

“…The same strategy was applied in [6] to find the structured eigenvalue backward error η S w (P, λ) for Hermitian and related structures. But the reformulation was aided by the fact that η…”

Section: Preliminariesmentioning

confidence: 99%

“…(For details, we refer to the proof of [6,Theorem 4.4].) However, as (3.2) suggests, this is not the case for the structured eigenvalue backward error η pal w (P, λ) for the -palindromic structures, because the function f in the right-hand side of (3.2) involves taking a maximum.…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Palindromic Structures

Bora¹,

Karow²,

Mehl³

et al. 2015

SIAM J. Matrix Anal. & Appl.

Self Cite

View full text Add to dashboard Cite

Abstract. We derive formulas for the backward error of an approximate eigenvalue of a * -palindromic matrix polynomial with respect to * -palindromic perturbations. Such formulas are also obtained for complex T -palindromic pencils and quadratic polynomials. When the T -palindromic polynomial is real, then we derive the backward error of a real number considered as an approximate eigenvalue of the matrix polynomial with respect to real T -palindromic perturbations. In all cases the corresponding minimal structure preserving perturbations are obtained as well. The results are illustrated by numerical experiments. These show that there is a significant difference between the backward errors with respect to structure preserving and arbitrary perturbations in many cases.Key words. palindromic matrix pencil, palindromic matrix polynomial, perturbation theory, eigenvalue backward error, structured eigenvalue backward error AMS subject classifications. 15A22, 15A18, 47A56, 15A60, 65F15, 65F30, 93C73 DOI. 10.1137/140973839 1. Introduction. Given n × n matrices A 0 , . . . , A m , the corresponding matrix polynomial P (z) :We consider such eigenvalue problems for the special case that the coefficient matrices of P (z) satisfy certain symmetries. This is indicated by stating that the ordered tuple (A 0 , . . . , A m ) belongs to S, where S ⊂ (C n×n ) m+1 . In particular, given λ ∈ C, we are interested in perturbations (Δ 0 , . . . , Δ m ) ∈ S to (A 0 , . . . , A m ) ∈ S that are minimal with respect to a specified norm such that λ is an eigenvalue of the perturbed polynomialThe norm of such a minimal structure preserving perturbation is called the structured backward error of λ as an approximate eigenvalue of P (z). We refer to this also as the structured eigenvalue backward error of λ with respect to P (z) and S. In contrast, we refer to the norm of a minimal but not necessarily structure preserving perturbation to P (z) such that λ ∈ C is an eigenvalue of the perturbed polynomial simply as the eigenvalue backward error of λ with respect to P (z).Matrix polynomials with symmetries in their coefficients are referred to as structured matrix polynomials. For example, the coefficients of Hermitian matrix polynomials are all Hermitian matrices, i.e., they satisfy, A

show abstract

Pseudospectra, stability radii and their relationship with backward error for structured nonlinear eigenvlaue problems

Ahmad,

Nag

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper discusses pseudospectra and stability radii for structured nonlinear matrix functions, such as Hermitian, skew‐Hermitian, H‐even, H‐odd, complex symmetric, and complex skew‐symmetric. To compute pseudospectra and stability radii, eigenvalue backward error is required. Hence, we initially present the structured eigenvalue backward error. Subsequently, we compute the structured pseudospectra using the obtained results for the eigenvalue backward error of a class of structured nonlinear matrix functions. Finally, we discuss the stability radii of the above‐structured problems arising in different applications. The paper also generalizes the results on the eigenvalue backward error of matrix polynomials in the literature for the above structures.

show abstract

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures

Cited by 19 publications

References 23 publications

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Polynomial Eigenvalue Problems: Theory, Computation, and Structure

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Palindromic Structures

Pseudospectra, stability radii and their relationship with backward error for structured nonlinear eigenvlaue problems

Contact Info

Product

Resources

About