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

Abstract

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

Structured backward error analysis of linearized structured polynomial eigenvalue problems

“…Only precise "local" structured backward error analyses valid for each particular computed eigenvalue or eigenpair have been developed so far. See, for example, [3,9,10,11], or [1,2] for the case of the structured linearizations in the vector spaces L 1 (P ), L 2 (P ) and DL(P ), introduced in [51,52] and [38].…”

Backward error analysis of specified eigenpairs for sparse matrix polynomials

“…Matrix polynomials with special structures occur in numerous applications in mechanics, control theory, linear systems theory and computer‐aided graphic design, see References 9,10,15,18. In particular, palindromic matrix polynomials arise in the mathematical modeling and numerical simulation of surface acoustic wave filters and vibration analysis of railway tracks excited by high‐speed trains, see References 11 and 3. For obtaining the eigenvalues and eigenvectors of matrix polynomials, the most widely used approach is to linearize the given matrix polynomial into a bigger size matrix pencil (see, Reference 13 for more on linearization).…”

Structural Stability of Matrix Pencils and Matrix Pairs Under Contragredient Equivalence

“…The concept of structural stability, first introduced by A. A. Andronov and L. S. Pontryagin in 1937 in the qualitative theory of dynamical systems, in the sense of structurally stable elements being those whose behavior does not change when applying small perturbation [1], has been largely studied by many authors in different contexts for example, in the scenario of Linear Algebra we can found [4], [5], [7], [8], among others.…”