Inequalities and bounds for the zeros of polynomials using Perron-Frobenius and Gerschgorin theories

In this paper, disks containing some, or all zeros of a complex polynomial or eigenvalues of a complex matrix are developed. These disks are based on extensions of Cauchy classical bounds, Perron-Probenius theory of positive matrices, and Gerschgorin theory. As a special case, given a real polynomial with real maximum or minimum zero, intervals containing the extreme zeros are developed. Moreover, methods for computing or refining these intervals are derived. Additionally, a closed form singular value decomposition of a characteristic polynomial was derived and utilized to compute new bounds for the zeros of polynomials. Finally, bounds that are based on zero transformation are given.