Structured backward error analysis of linearized structured polynomial eigenvalue problems

Abstract

We start by introducing a new class of structured matrix polynomials, namely, the class of M A -structured matrix polynomials, to provide a common framework for many classes of structured matrix polynomials that are important in applications: the classes of (skew-)symmetric, (anti-)palindromic, and alternating matrix polynomials. Then, we introduce the families of M Astructured strong block minimal bases pencils and of M A -structured block Kronecker pencils, which are particular examples of block minimal bases pencils recently introduced by Dopico, Lawrence, Pérez and Van Dooren, and show that any M A -structured odd-degree matrix polynomial can be strongly linearized via an M A -structured block Kronecker pencil. Finally, for the classes of (skew-)symmetric, (anti-)palindromic, and alternating odd-degree matrix polynomials, the M Astructured framework allows us to perform a global and structured backward stability analysis of complete structured polynomial eigenproblems, regular or singular, solved by applying to a M Astructured block Kronecker pencil a structurally backward stable algorithm that computes its complete eigenstructure, like the palindromic-QR algorithm or the structured versions of the staircase algorithm. This analysis allows us to identify those M A -structured block Kronecker pencils that yield a computed complete eigenstructure which is the exact one of a slightly perturbed structured matrix polynomial. These pencils include (modulo permutations) the well-known block-tridiagonal and block-antitridiagonal structure-preserving linearizations. Our analysis incorporates structure to the recent (unstructured) backward error analysis performed for block Kronecker linearizations by Dopico, Lawrence, Pérez and Van Dooren, and share with it its key features, namely, it is a rigorous analysis valid for finite perturbations, i.e., it is not a first order analysis, it provides precise bounds, and it is valid simultaneously for a large class of structure-preserving strong linearizations.

Block Kronecker linearizations of matrix polynomials and their backward errors

“…The results in this work have already motivated considerable research in the area. For instance, they have clarified many of the results that have been published in the last few years on linearizations of matrix polynomials, since it has been proved in [10] that all generalized Fiedler linearizations [3,8,15], all Fiedler linearizations with repetition [7,11,74], and all generalized Fiedler linearizations with repetition [9] may be transformed through proper permutations into particular strong block minimal bases pencils that can be described very easily; structured versions of the backward error analysis in this paper have been developed for many classes of structured strong block minimal bases linearizations of structured matrix polynomials in [26]; in [62] particular block minimal bases linearizations have been used to compute efficiently and in a stable way the zeros of a polynomial that is the sum of two polynomials expressed in two different bases, as well as for solving other challenging numerical problems; extensions of block Kronecker pencils that linearize matrix polynomials expressed in Chebyshev bases have been developed in [48]; it has been shown that each strong block minimal bases pencil can be used to construct strong linearizations of rational matrices with non-constant polynomial part [2]; etc. In addition to these publications, several other ongoing research projects related to block minimal bases pencils are being currently developed by different researchers.…”

Strong linearizations of rational matrices with polynomial part expressed in an orthogonal basis

“…The approach in Example 7.1 can be extended to other symmetric strong block minimal bases pencils of the symmetric polynomial part D(λ) of G(λ) = G(λ) T to construct other symmetric strong linearizations of G(λ), as long as D(λ) has odd-degree. See, for instance, the pencils considered in [14]. However, the linearization in Example 7.1 is particularly simple and, in view of the results in [7], we expect that it will have favourable numerical properties.…”

A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error

“…For the sake of brevity, we omit a detailed list of the many references published recently on linearizations of matrix polynomials and, instead, we invite the reader to check the references included in [10,14,43]. Unfortunately, this explosion of new classes of linearizations has not been followed by the corresponding analyses of their numerical properties, i.e., by the study of the errors they produce when they are used for solving numerically a PEP, and the number of papers analyzing this question is still low [1,44,14,15,23,24,39,45].…”