On the Estimation of the Numerical Range of Monic Matrix Polynomials (original) (raw)

The boundary of the numerical range of matrix polynomials

Linear Algebra and its Applications, 1997

Some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper. We also consider isolated points of the numerical range and the location of the numerical range in a circular annulus. 0 1997 Elsevier Science Inc.

On the connectedness of numerical range of matrix polynomials

Linear Algebra and its Applications, 1998

An investigation on nonconnectedness of numerical range for manic matrix polynomials L(1) is undertaking here. The distribution of eigenvalues of L(1) to the components of numerical range and some other algebraic properties are also presented.

Polytope-based computation of polynomial ranges

Computer Aided Geometric Design, 2012

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate a polytope defined as the convex hull of the coefficients with respect to the tensorial Bernstein basis, and we formulate several polytopes based on the Bernstein polynomials of the domain. These Bernstein polytopes can be defined by a polynomial number of halfspaces. We give the number of vertices, the number of hyperfaces, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 10.

Generalization of numerical range of polynomial operator matrices

Tikrit Journal of Pure Science

Suppose that is a polynomial matrix operator where for , are complex matrix and let be a complex variable. For an Hermitian matrix , we define the -numerical range of polynomial matrix of as , where . In this paper we study and our emphasis is on the geometrical properties of . We consider the location of in the complex plane and a theorem concerning the boundary of is also obtained. Possible generalazations of our results including their extensions to bounded linerar operators on an infinite dimensional Hilbert space are described.

Estimations of zeros of a polynomial using numerical radius inequalities

2020

We present new bounds for the numerical radius of bounded linear operators and 2times22\times 22times2 operator matrices. We apply upper bounds for the numerical radius to the Frobenius companion matrix of a complex monic polynomial to obtain new estimations for zeros of that polynomial. We also show with numerical examples that our new estimations improve on the existing estimations.

Polynomial numerical hulls of matrices

Linear Algebra and its Applications, 2008

For any n-by-n complex matrix A, we use the joint numerical range W (A, A 2 , . . . , A k ) to study the polynomial numerical hull of order k of A, denoted by V k (A). We give an analytic description of V 2 (A) when A is normal. The result is then used to characterize those normal matrices A satisfying V 2 (A) = σ(A), and to show that a unitary matrix A satisfies V 2 (A) = σ(A) if and only if its eigenvalues lie in a semicircle, where σ(A) denotes the spectrum of A. When A = diag (1, w, . . . , w n−1 ) with w = e i2π/n , we determine V k (A) for k ∈ {2}∪{j ∈ N : j ≥ n/2}.