Locating the Eigenvalues of Matrix Polynomials

Бини, Дарио Андреа; Noferini, Vanni; Sharify, Meisam

doi:10.1137/120886741

Cited by 50 publications

(52 citation statements)

References 18 publications

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“…Simple location rules are used for theoretical purposes, as establishing sufficient conditions guaranteeing that p(z) is stable or that all its roots are inside the unit circle, and they are also used in iterative algorithms for computing the roots of p(z) to find initial guesses of the roots for starting the iteration [2,3]. Recently, polynomial eigenvalue problems have received much attention and simple criteria for locating approximately the eigenvalues of matrix polynomials have been developed [4,12], but, to keep the paper concise, matrix polynomials are not covered in this work. Let us denote by λ any root of p(z).…”

Section: Introductionmentioning

confidence: 99%

New bounds for roots of polynomials based on Fiedler companion matrices

Terán

Dopico

Pérez

2014

Linear Algebra and its Applications

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Several matrix norms of the classical Frobenius companion matrices of a monic polynomial p(z) have been used in the literature to obtain simple lower and upper bounds on the absolute values of the roots λ of p(z). Recently, M. Fiedler has introduced a new family of companion matrices of p(z) (Lin. Alg. Appl., 372 (2003) 325-331) that has received considerable attention and it is natural to investigate if matrix norms of Fiedler companion matrices may be used to obtain new and sharper lower and upper bounds on |λ|. The development of such bounds requires first to know simple expressions for some relevant matrix norms of Fiedler matrices and we obtain them in the case of the 1-and ∞-matrix norms. With these expressions at hand, we will show that norms of Fiedler matrices produce many new bounds, but that none of them improves significatively the classical bounds obtained from the Frobenius companion matrices. However, we will prove that if the norms of the inverses of Fiedler matrices are used, then another family of new bounds on |λ| is obtained and some of the bounds in this family improve significatively the bounds coming from the Frobenius companion matrices for certain polynomials.

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Section: Introductionmentioning

confidence: 99%

New bounds for roots of polynomials based on Fiedler companion matrices

Terán

Dopico

Pérez

2014

Linear Algebra and its Applications

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“…For the particular case of matrix polynomials P (λ) with coefficient matrices of the form A i = σ i Q i with σ i ≥ 0 and Q * i Q i = I, and the 2-norm · 2 , Bini, Noferini, and Sharify [7,Thm. 2.7] have identified annuli of small width defined in terms of the tropical roots of t × p(x) that contain the eigenvalues of P (λ).…”

Section: −1mentioning

confidence: 99%

“…is useful in a number of situations, such as, for example, when selecting the starting points in the Ehrlich-Aberth method for the numerical solution of polynomial eigenvalue problems [6], [7], or in choosing the contour in contour integral methods for polynomial eigenvalue problems of large dimensions [3]. Betcke's diagonal scaling [4, section 5], whose aim is to improve the conditioning of P 's eigenvalues near a target eigenvalue ω, requires a priori knowledge of the magnitude of ω.…”

Section: Introductionmentioning

confidence: 99%

Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials

Noferini¹,

Sharify²,

Tisseur³

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. The tropical roots of t × p(x) = max 0≤i≤ A i x i are points at which the maximum is attained for at least two values of i for some x. These roots, which can be computed in only O( ) operations, can be good approximations to the moduli of the eigenvalues of the matrix polynomial P (λ) = i=0 λ i A i , in particular when the norms of the matrices A i vary widely. Our aim is to investigate this observation and its applications. We start by providing annuli defined in terms of the tropical roots of t × p(x) that contain the eigenvalues of P (λ). Our localization results yield conditions under which tropical roots offer order of magnitude approximations to the moduli of the eigenvalues of P (λ). Our tropical localization of eigenvalues is less tight than eigenvalue localization results derived from a generalized matrix version of Pellet's theorem but they are easier to interpret. Tropical roots are already used to determine the starting points for matrix polynomial eigensolvers based on scalar polynomial root solvers such as the Ehrlich-Aberth method and our results further justify this choice. Our results provide the basis for analyzing the effect of Gaubert and Sharify's tropical scalings for P (λ) on (a) the conditioning of linearizations of tropically scaled P (λ) and (b) the backward stability of eigensolvers based on linearizations of tropically scaled P (λ). We anticipate that the tropical roots of t × p(x), on which the tropical scalings are based will help designing polynomial eigensolvers with better numerical properties than standard algorithms for polynomial eigenvalue problems such as that implemented in the MATLAB function polyeig.

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“…We can give reasonable estimates to the modulus of the eigenvalues using the Pellet theorem or the tropical roots (see [19,46,64], for some insight on these tools). We have considered three linearizations: the standard Frobenius companion matrix, and two versions of the extended Smith companion form.…”

Section: The Matrix Casementioning

confidence: 99%