Bounding the Zeros of Polynomials Using the Frobenius Companion Matrix Partitioned by the Cartesian Decomposition

Abstract

In this work, some new inequalities for the numerical radius of block n-by-n matrices are presented. As an application, the bounding of zeros of polynomials using the Frobenius companion matrix partitioned by the Cartesian decomposition method is proved. We highlight several numerical examples showing that our approach to bounding zeros of polynomials could be very effective in comparison with the most famous results as well as some recent results presented in the field. Finally, observations, a discussion, and a conclusion regarding our proposed bound of zeros are considered. Namely, it is proved that our proposed bound is more efficient than any other bound under some conditions; this is supported with many polynomial examples explaining our choice of restrictions.