Generalization and variations of Pellet’s theorem for matrix polynomials

Melman, Aaron

doi:10.1016/j.laa.2013.05.003

Cited by 31 publications

(29 citation statements)

References 15 publications

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“…[17] the result can be extended to any subordinate norm by using Theorem 2.1. Throughout this section, roots of polynomials are counted with multiplicity: in particular, a double positive root is thought of as two coincident positive roots.…”

Section: It Does Not Have Any Eigenvalue In the Open Annulusmentioning

confidence: 98%

“…As shown in section 3 our tropical localization results are less tight than those from the generalized Pellet's theorem, both in the form given in [7,Thm. 2.1] and in [17,Thm. 3.3], but they are easier to interpret.…”

Section: −1mentioning

confidence: 99%

“…Our aim is to show that, under specific conditions, the tropical roots of t × p(x) are good approximations to the moduli of the eigenvalues of P (λ). The key tool for this is a generalization of Rouché's theorem for matrix valued functions [10], [17]. Before deriving our new results, we set up the notation used throughout the reminder of this paper.…”

Section: Eigenvalue Location: Tropical Approachmentioning

confidence: 99%

“…The eigenvalues of P (λ), which we computed with the MATLAB function polyeig, are located in three separate annuli A i , i = 1, 2, 3, given in the first rows of Table 2. These are to be compared to the annuli from the Bini et al generalized Pellet's theorem (see Theorem 3.1), Melman's generalized Pellet's theorem (see [17,Thm. 3.3] or Theorem 3.1 with q k (x) in place of q k (x)), and that of Theorem 2.7 referred to as Pellet 1, Pellet 2, and Tropical, respectively, in Table 2.…”

Section: Numerical Experiments and Applicationsmentioning

confidence: 99%

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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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Section: It Does Not Have Any Eigenvalue In the Open Annulusmentioning

confidence: 98%

Section: −1mentioning

confidence: 99%

Section: Eigenvalue Location: Tropical Approachmentioning

confidence: 99%

Section: Numerical Experiments and Applicationsmentioning

confidence: 99%

See 2 more Smart Citations

Tropical Roots as Approximations to Eigenvalues of Matrix Polynomials

Noferini¹,

Sharify²,

Tisseur³

2015

SIAM J. Matrix Anal. & Appl.

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show abstract

“…The matrix norms are assumed to be subordinate (induced by a vector norm). [2,7,15].) All eigenvalues of the matrix polynomial P (z) = A n z n + A n−1 z n−1 + · · · + A 1 z + A 0 , where n ≥ 2 and A j ∈ C m×m , for j = 0, .…”

Section: Theorem 41 (Cauchy's Theorem -Original Scalar Version) (Seementioning

confidence: 99%