Containment regions for zeros of polynomials from numerical ranges of companion matrices

Containment Regions for Zeros of Analytic Functions

“…The paper extends some of the results of [15,16] for polynomials to the case of analytic and entire functions. In addition, the method is extended to certain meromorphic functions.…”

“…Lei the assumptions (16) and (17) be satisfied. Then the function z f{z) := 1 -a*:^* is holomorphic in {z : \z\ < where ß is given by (19).…”

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.…”