Block minimal bases ℓ-ifications of matrix polynomials

Dopico, Froilán M.; Pérez, Javier; Dooren, Paul Van

doi:10.1016/j.laa.2018.10.010

Cited by 12 publications

(41 citation statements)

References 52 publications

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“…But, first, we notice that if we do not impose the M Astructure on the pencil M (λ), it is possible to prove that the polynomial equation (4.4) is always consistent, with infinitely many solutions, as a consequence of the properties of the minimal basis N (λ). This result will be proved in [27], in a much more general setting.…”

Section: Strong Block Minimal Bases Pencils and Mmentioning

confidence: 77%

Structured backward error analysis of linearized structured polynomial eigenvalue problems

2018

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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.

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Section: Strong Block Minimal Bases Pencils and Mmentioning

confidence: 77%

Structured backward error analysis of linearized structured polynomial eigenvalue problems

2018

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“…In this paper it is enough to know that an ℓ-ification of a polynomial matrix P (λ) is another polynomial matrix of degree ℓ that has the same (finite and infinite) elementary divisors, the same number of left, and the same number of right minimal indices as P (λ). The results in [11] pose naturally the question whether or not the backward error analysis for block Kronecker linearizations in [10,Section 6] can be extended to the classes of ℓ-ifications introduced in [11]. This seems to be a hard problem whose potential solution will be related for sure to the results in this paper.…”

Section: Applications To Backward Error Analysesmentioning

confidence: 84%

“…A second application is related to the fact that the SBMB and block Kronecker linearizations of polynomial matrices recently introduced in [10] can be extended to SBMB and block Kronecker ℓ-ifications of polynomial matrices [11], which are constructed by using polynomial matrices with full-Sylvester-rank of degree ℓ and whose right minimal indices are all equal. The reader is referred to [6,9] for formal definitions of the concept of an ℓ-ification of a polynomial matrix.…”

Section: Applications To Backward Error Analysesmentioning

confidence: 99%

“…Very recently, minimal bases, and the closely related notion of pairs of dual minimal bases, have been applied to the solution of some problems that have attracted the attention of many researchers in the last fifteen years. For instance, minimal bases have been used (1) in the solution of inverse complete eigenstructure problems for polynomial matrices (see [7,8] and the references therein), (2) in the development of new classes of linearizations and ℓ-ifications of polynomial matrices [5,9,10,11,12,18,22], which has allowed to recognize that many important linearizations commonly used in the literature are constructed via dual minimal bases (including the classical Frobenius companion forms), (3) in the explicit construction of linearizations of rational matrices [1], and (4) in the backward error analysis of complete polynomial eigenvalue problems solved via the so-called "block Kronecker linearizations" of polynomial matrices [10,Section 6], which include the interesting class of Fiedler linearizations (see [5,10] for references on this class of linearizations), but do not include most of the linearizations and ℓ-ifications that can be constructed from minimal bases [5,9,10,11,12,18,22]. See also [19] for additional references on the role played by dual minimal bases in backward error analyses.…”

Section: Introductionmentioning

confidence: 99%

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Robustness and perturbations of minimal bases

Dooren

Dopico

2018

Linear Algebra and its Applications

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Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory, linear systems theory, and coding theory. It is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis. Very recently, minimal bases, as well as the closely related pairs of dual minimal bases, have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices, the development of new classes of linearizations and ℓ-ifications of polynomial matrices, and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations. These new applications have revealed that although the algebraic properties of minimal bases are rather well understood, their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature, as far as we know. Therefore, the main purpose of this paper is to study in detail when a minimal basis M (λ) is robust under perturbations, i.e., when all the polynomial matrices in a neighborhood of M (λ) are minimal bases, and, in this case, how perturbations of M (λ) change its dual minimal bases. In order to study such problems, a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and, based on it, we prove that polynomial matrices are generically minimal bases with very specific properties. In addition, some applications of the results of this paper are discussed.

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“…Indeed there is much current research into what is called " -ification," i.e. reduction of a matrix polynomial of degree m to a (larger) matrix polynomial of degree at most (having degree at most is also called "having grade ") [1]. But here we restrict ourselves to the case = 1.…”

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confidence: 99%

Algebraic linearizations of matrix polynomials

Chan

Corless

González-Vega

et al. 2019

Linear Algebra and its Applications

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We show how to construct linearizations of matrix polynomials za(z)d 0 + c 0 ,and za(z)d 0 b(z)+c 0 from linearizations of the component parts, a(z) and b(z). This allows the extension to matrix polynomials of a new companion matrix construction. Keywords: companion matrices, linearization, matrix polynomials, block upper Hessenberg 2010 MSC: 65F99, 15A22 IntroductionMany applications require the computation or approximation of polynomial eigenvalues, that is, those z ∈ C for which the matrix polynomial P(z) (of degree at most s) ∈ C[z] r×r is singular. In other words, we search for z such that det P(z) = 0. If s = 1, that is P(z) = zB−A, where A, B ∈ C N ×N , where N = r, is degree 1 in z, i.e. linear, then this is "just" the generalized eigenvalue problem, which can be reliably solved numerically on many platforms using software developed over many decades by the efforts of many people. We do

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Block minimal bases ℓ-ifications of matrix polynomials

Cited by 12 publications

References 52 publications

Structured backward error analysis of linearized structured polynomial eigenvalue problems

Structured backward error analysis of linearized structured polynomial eigenvalue problems

Robustness and perturbations of minimal bases

Algebraic linearizations of matrix polynomials

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