Abstract. In many applications, the polynomial eigenvalue problem, P (λ)x = 0, arises with P (λ) being a structured matrix polynomial (for example, palindromic, symmetric, skew-symmetric). In order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as P (λ) to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations. In this paper we characterize all the pencils in the family of the Fiedler pencils with repetition, introduced by Vologiannidis and Antoniou [25], associated with a square matrix polynomial P (λ) that are block-symmetric for every matrix polynomial P (λ). We show that this family of pencils is precisely the set of all Fiedler pencils with repetition that are symmetric when P (λ) is. When some generic nonsingularity conditions are satisfied, these pencils are strong linearizations of P (λ). In particular, our family strictly contains the standard basis for DL(P ), a k-dimensional vector space of symmetric pencils introduced by Mackey, Mackey, Mehl, and Mehrmann [20].
Abstract. Given a matrix polynomial P (λ) = P k i=0 λ i A i of degree k, where A i are n × n matrices with entries in a field F, the development of linearizations of P (λ) that preserve whatever structure P (λ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P (λ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P (λ) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space called DL(P ) of dimension k of block-symmetric pencils such that most of them are linearizations, which form an infinite family when the field F is infinite. One drawback of the pencils in DL(P ) is that none of them is a linearization when P (λ) is singular. In this paper we introduce many new classes of block-symmetric linearizations for P (λ), when k ≥ 3, which have three key features: (a) their constructions are based on an new class of Fiedler-like pencils that we call generalized Fiedler pencils with repetition; (b) they are defined in terms of vector spaces of dimension O(n 2 ) of block-symmetric pencils most of which are linearizations; and (c) when k is odd, many of these classes contain linearizations even when P (λ) is singular. Therefore, the two fundamental contributions of this manuscript are that the dimensions of the new spaces of block-symmetric linearizations are much larger than the dimension of DL(P ) (for n ≥ √ k) and that vector spaces of block-symmetric linearizations valid for singular matrix polynomials are presented for the first time.In particular, the largest dimension of the subspaces of block-symmetric pencils we introduce is jand the coefficients of the pencils in the new subspaces can be easily constructed as k × k block-matrices whose n × n blocks are of the form 0, αIn, ±αA i , or arbitrary n × n matrices, where α is an arbitrary nonzero scalar.
A standard way to solve polynomial eigenvalue problems P (λ)x = 0 is to convert the matrix polynomial P (λ) into a matrix pencil that preserves its elementary divisors and, therefore, its eigenvalues. This process is known as linearization and is not unique, since there are infinitely many linearizations with widely varying properties associated with P (λ). This freedom has motivated the recent development and analysis of new classes of linearizations that generalize the classical first and second Frobenius companion forms, with the goals of finding linearizations that retain whatever structures that P (λ) might possess and/or of improving numerical properties, as conditioning or backward errors, with respect the companion forms. In this context, an important new class of linearizations is what we name generalized Fiedler linearizations, introduced in 2004 by Antoniou and Vologiannidis as an extension of certain linearizations introduced previously by Fiedler for scalar polynomials. On the other hand, the mere definition of linearization does not imply the existence of simple relationships between the eigenvectors, minimal indices, and minimal bases of P (λ) and those of the linearization. So, given a class of linearizations, to provide easy recovery procedures for eigenvectors, minimal indices, and minimal bases of P (λ) from those of the linearizations is essential for the usefulness of this class. In this paper we develop such recovery procedures for generalized Fiedler linearizations and pay special attention to structure preserving linearizations inside this class.
The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to embed the matrix polynomial into a matrix pencil, transforming the problem into an equivalent generalized eigenvalue problem. Such pencils are known as linearizations. Many of the families of linearizations for matrix polynomials available in the literature are extensions of the so-called family of Fiedler pencils. These families are known as generalized Fiedler pencils, Fiedler pencils with repetition and generalized Fiedler pencils with repetition, or Fiedler-like pencils for simplicity. The goal of this work is to unify the Fiedler-like pencils approach with the more recent one based on strong block minimal bases pencils introduced in [15]. To this end, we introduce a family of pencils that we have named extended block Kronecker pencils, whose members are, under some generic nonsingularity conditions, strong block minimal bases pencils, and show that, with the exception of the non proper generalized Fiedler pencils, all Fiedler-like pencils belong to this family modulo permutations. As a consequence of this result, we obtain a much simpler theory for Fiedler-like pencils than the one available so far. Moreover, we expect this unification to allow for further developments in the theory of Fiedler-like pencils such as global or local backward error analyses and eigenvalue conditioning analyses of polynomial eigenvalue problems solved via Fiedler-like linearizations.Key words. Fiedler pencils, generalized Fiedler pencils, Fiedler pencils with repetition, generalized Fiedler pencils with repetition, matrix polynomials, strong linearizations, block minimal bases pencils, block Kronecker pencils, extended block Kronecker pencils, minimal basis, dual minimal bases AMS subject classifications. 65F15, 15A18, 15A22, 15A54 1. Introduction. Matrix polynomials and their associated polynomial eigenvalue problems appear in many areas of applied mathematics, and they have received in the last years considerable attention. For example, they are ubiquitous in a wide range of problems in engineering, mechanic, control theory, computer-aided graphic design, etc. For detailed discussions of different applications of matrix polynomials, we refer the reader to the classical references [23,28,42], the modern surveys [2, Chapter 12] and [37,43] (and their references therein), and the references [32,33,34]. For those readers not familiar with the theory of matrix polynomials and polynomial eigenvalue problems, those topics are briefly reviewed in Section 2.The standard way of solving the polynomial eigenvalue problem associated with a matrix polynomial is to linearize the polynomial into a matrix pencil (i.e., matrix polynomials of grade 1), known as linearization [13,22,23]. The linearization process transforms the polynomial eigenvalue problem into an equivalent generalized eigenvalue problem, which, then, can be solved using mature and well-understood eigensolvers such as the QZ algorithm or the
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.