Numerical Radii of Some Companion Matrices and Bounds for the Zeros of Polynomials

Bounds for the zeros of polynomials from matrix inequalities

“…This is a continuation of our earlier work [10] and [13]. See, also, [3], [7], [14], [15], [16], and references therein for bounds of the zeros of polynomials obtained from applying matrix inequalities to various types of companion matrices of monic polynomials. These inequalities involve the Gersgorin theorem, basic eigenvalue-singular value majorization relations, matrix norms computations, and numerical radii estimations.…”

“…A comparison between these bounds has been also given in [15], and generalizations of them have been given there by considering generalized companion matrices instead of the Frobenius companion matrix.…”

Generalizations of Gershgorin disks and polynomial zeros

“…We concentrate on this frequently used companion matrix, but there exist other companion matrices that could also be used, as in, e.g., [2], [6], [7], [14], [18], [25], [26], [27], or [36]. In the following theorem we derive an inclusion set for polynomial zeros, based on Theorem 2.1.…”

“…316-319]), there exist several more modern methods, such as in [1], [5], [11], [12], [15], [20], [21], [22], [25], [26], [27], [28], [34], [37], [38], which were already mentioned in the introduction. Judging from the numerical examples in those references, these bounds are, by and large, comparable.…”

“…The literature on the computation of polynomial zeros and bounds on such zeros is extensive, and we refer to [1], [4], [5], [6], [11], [12], [13], [15], [16], [20], [21], [22], [25], [26], [27], [28], [31], [32], [34], [37], [38], and the references therein, to name but a few. Most of these take a linear algebra approach, but some do not.…”