Geometry of Polynomials

Marden, Morris

doi:10.1090/surv/003

Cited by 787 publications

(622 citation statements)

References 48 publications

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“…. , , be the partial sums defined in (15), and let Mσ(p) be the Fiedler matrix of p(z) associated with σ. Let us define the quantities…”

Section: Definition and Basic Properties Of Fiedler Matricesmentioning

confidence: 99%

“…To locate approximately the roots of p(z) through simple operations with its coefficients is a classical problem that has produced a considerable amount of literature (see the comprehensive surveys [15,17] and the references therein). 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].…”

Section: Introductionmentioning

confidence: 99%

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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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“…. , , be the partial sums defined in (15), and let Mσ(p) be the Fiedler matrix of p(z) associated with σ. Let us define the quantities…”

Section: Definition and Basic Properties Of Fiedler Matricesmentioning

confidence: 99%

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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“…These bounds are tied to the well-known estimates for polynomial roots by Cauchy, Lagrange, and others (see, for example, [19]). Yet another proof of this result appears in [20], along with some extensions, based on using Birkhoff-James orthogonality more directly.…”

Section: An Application To Zeros Of Analytic Functionsmentioning

confidence: 99%

Multipliers of sequence spaces

2017

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“…Some of the intricacies of the original approach are eased by an inductive use of the following famous result about polynomials (for which a good reference is [50]). …”

Section: Stable Polynomialsmentioning

confidence: 99%

The mathematics of eigenvalue optimization

Lewis

2003

Math. Program., Ser. B

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Abstract. Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for the broad optimization community. I discuss the convex analysis of spectral functions and invariant matrix norms, touching briefly on semidefinite representability, and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and Lie algebra. Analogous nonconvex notions lead into eigenvalue perturbation theory. The last third of the article concerns stability, for polynomials, matrices, and associated dynamical systems, ending with a section on robustness. The powerful and elegant language of nonsmooth analysis appears throughout, as a unifying narrative thread.Key words. Eigenvalue optimization -convexity -nonsmooth analysis -duality -semidefinite program -subdifferential -Clarke regular -chain rule -sensitivity -eigenvalue perturbation -partly smooth -spectral function -unitarily invariant norm -hyperbolic polynomial -stability -robust control -pseudospectrum -H∞ norm PART I: INTRODUCTION Von Neumann and invariant matrix normsBefore outlining this survey, I would like to suggest its flavour with a celebrated classical result. This result, von Neumann's characterization of unitarily invariant matrix norms, serves both as a historical jumping-off point and as an elegant juxtaposition of the central ingredients of this article.Von Neumann [64] was interested in unitarily invariant norms · on the vector space M n of n-by-n complex matrices:where U n denotes the group of unitary matrices. The singular value decomposition shows that the invariants of a matrix X under unitary transformations of the form X → U XV are given by the singular values σ 1 (X) ≥ σ 2 (X) ≥ · · · ≥ σ n (X), the eigenvalues of the matrix √ X * X. Hence any invariant norm · must be a function of the vector σ(X), so we can write X = g(σ(X)) for some function g : R n → R. We ensure this last equation if we define g(x) = Diag x for vectors x ∈ R n , and in that case g is a symmetric gauge function: that is, g is a norm on R n whose value is invariant under permutations and sign changes of the components. Von Neumann's beautiful insight was that this simple necessary condition is in fact also sufficient.Theorem 1.1 (von Neumann, 1937). The unitarily invariant matrix norms are exactly the symmetric gauge functions of the singular values.Von Neumann's proof rivals the result in elegance. He proceeds by calculating the dual norm of a matrix Y ∈ M n ,where the real inner product X, Y in M n is the real part of the trace of X * Y . Specifically, he shows that any symmetric gauge function g satisfies the simple duality relationshipwhere g * is the norm on R n dual to g. From this, the hard part of his result follows:...

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Geometry of Polynomials

Cited by 787 publications

References 48 publications

New bounds for roots of polynomials based on Fiedler companion matrices

New bounds for roots of polynomials based on Fiedler companion matrices

Multipliers of sequence spaces

The mathematics of eigenvalue optimization

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