Estimation of bounds for the zeros of a polynomial using numerical radius

The estimation of zeros of a polynomial have been done by many mathematicians over the years using various approaches. In this paper we estimate the upper bound for the zeros of a given polynomial using Hilbert space technique involving Frobenius companion matrix and numerical radius. We first obtain numerical range and numerical radius for certain class of matrices and use them to estimate the bounds for zeros of a given polynomial. We illustrate with examples to show that the estimations obtained here is better than the previously known estimations. We also obtain a sequence of real numbers which converges exactly to the spectral radius of some special class of matrices.