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

Bueno, M.I.; Dopico, Froilán M.; Furtado, Susana; Medina, Luis

doi:10.1007/s10092-018-0273-4

Cited by 13 publications

(14 citation statements)

References 41 publications

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“…One can obtain tighter constants, for example, particularizing the analysis to block Kronecker companion forms such that the matrices M i are of low blockbandwidth. In this case, the constant reduces essentially to d 3 , result that is coherent with other analyses; see [5,Theorem 5.1].…”

Section: 2supporting

confidence: 85%

The conditioning of block Kronecker $\ell$-ifications of matrix polynomials

Pérez

2018

Preprint

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A strong ℓ-ification of a matrix polynomial P (λ) = A i λ i of degree d is a matrix polynomial L(λ) of degree ℓ having the same finite and infinite elementary divisors, and the same number of left and right minimal indices as P (λ). Strong ℓ-ifications can be used to transform the polynomial eigenvalue problem associated with P (λ) into an equivalent polynomial eigenvalue problem associated with a larger matrix polynomial L(λ) of lower degree. Of most interest in applications is ℓ = 1, for which L(λ) receives the name of strong linearization. However, there exist some situations, e.g., the preservation of algebraic structures, in which it is more convenient to replace strong linearizations by other low degree matrix polynomials. In this work, we investigate the eigenvalue conditioning of ℓ-ifications from a family of matrix polynomials recently identified and studied by Dopico, Pérez and Van Dooren, the so-called block Kronecker companion forms. We compare the conditioning of these ℓ-ifications with that of the matrix polynomial P (λ), and show that they are about as well conditioned as the polynomial itself, provided we scale P (λ) so that max{ A i 2 } = 1, and the quantity min{ A 0 2 , A d 2 } is not too small. Moreover, under the scaling assumption max{ A i 2 } = 1, we show that any block Kronecker companion form, regardless of its degree or block structure, is about as well-conditioned as the well-known Frobenius companion forms. Our theory is illustrated by numerical examples.

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Section: 2supporting

confidence: 85%

The conditioning of block Kronecker $\ell$-ifications of matrix polynomials

Pérez

2018

Preprint

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“…Thus, ignoring again the factor Z k , the three conditions cond ∞ (A) ≈ 1, ρ ≈ 1, and ρ ≈ 1 are sufficient to imply that all the bounds in Theorem 5.8 are moderate numbers and guarantee that the Möbius transformation M A does not change significantly the relative eigenvalue condition numbers of any eigenvalue of a matrix polynomial P satisfying ρ ≈ 1 and ρ ≈ 1. Note that the presence of ρ and ρ is natural, since ρ has appeared previously in a number of results that compare the relative eigenvalue condition numbers of a matrix polynomial and of some of its linearizations [19,6].…”

Section: Q Rmentioning

confidence: 99%

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

2019

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Möbius transformations have been used in numerical algorithms for computing eigenvalues and invariant subspaces of structured generalized and polynomial eigenvalue problems (PEPs). These transformations convert problems with certain structures arising in applications into problems with other structures and whose eigenvalues and invariant subspaces are easily related to the ones of the original problem. Thus, an algorithm that is efficient and stable for some particular structure can be used for solving efficiently another type of structured problem via an adequate Möbius transformation. A key question in this context is whether these transformations may change significantly the conditioning of the problem and the backward errors of the computed solutions, since, in that case, their use might lead to unreliable results. We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of PEPs. By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that, if the matrix inducing the Möbius transformation is well conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the whole polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the Möbius transformations induced by well-conditioned matrices only if a penalty factor, depending on the norms of those matrix coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation of the PEP is used.

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“…The interest of the community in such constructions resulted in several families of structured linearizations (not companion) [4,38]. Later on, structured companion linearizations were obtained for the palindromic structure [14,19], for the symmetric structure [10,13,15,16], and for all of them [28] (note that some of these are quite recent references). However, these structured companion linearizations are not valid for all polynomials.…”

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confidence: 99%

“…13. Let P (λ) = 21 j=0 λ j P j ∈ F[λ] n×n be a -symmetric matrix polynomial of grade 21, and let X, Y ∈ F n×n be invertible matrices.…”

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confidence: 99%

Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis

Terán

Hernando

Pérez

2021

ELA

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In the framework of Polynomial Eigenvalue Problems (PEPs), most of the matrix polynomials arising in applications are structured polynomials (namely, (skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The standard way to solve PEPs is by means of linearizations. The most frequently used linearizations belong to general constructions, valid for all matrix polynomials of a fixed degree, known as companion linearizations. It is well known, however, that it is not possible to construct companion linearizations that preserve any of the previous structures for matrix polynomials of even degree. This motivates the search for more general companion forms, in particular companion $\ell$-ifications. In this paper, we present, for the first time, a family of (generalized) companion $\ell$-ifications that preserve any of these structures, for matrix polynomials of degree $k=(2d+1)\ell$. We also show how to construct sparse $\ell$-ifications within this family. Finally, we prove that there are no structured companion quadratifications for quartic matrix polynomials.

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

Abstract: A block-symmetric linearization of odd degree matrix polynomials with optimal eigenvalue condition number and backward error. Calcolo, 55(3).

Cited by 13 publications

References 41 publications

The conditioning of block Kronecker $\ell$-ifications of matrix polynomials

The conditioning of block Kronecker $\ell$-ifications of matrix polynomials

Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations

Structured strong $\boldsymbol{\ell}$-ifications for structured matrix polynomials in the monomial basis

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