Bounds for Polynomial’s Roots from Fiedler and Sparse Companion Matrices for Submultiplicative Matrix Norms

Abstract

We use submultiplicative companion matrix norms to provide new bounds for roots for a given polynomial P ( X ) over the field C[ X ]. From a n × n Fiedler companion matrix C , sparse companion matrices and triangular Hessenberg matrices are introduced. Then, we identify a special triangular Hessenberg matrix L r , supposed to provide a good estimation of the roots. By application of Gershgorin’s theorems to this special matrix in case of submultiplicative matrix norms, some estimations of bounds for roots are made. The obtained bounds have been compared to known ones from the literature precisely Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds. According to the starting formel of L r , we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.