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 Lr, 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 Lr, we see that the more we have coefficients closed to zero with a norm less than 1, the more the Sparse method is useful.
Highlights
Approximating of roots for polynomials has been subject of several studies in mathematics and physics especially stability studies of dynamic system and automatism
The methodology of providing a sharp bound for polynomial’s roots consists of identifying a special form of Hessenberg matrix that could give an improved value of bound in case of submultiplicative matrix norms
We compare bounds of roots of polynomial obtained by the Sparse method, to those found in the literature
Summary
Approximating of roots for polynomials has been subject of several studies in mathematics and physics especially stability studies of dynamic system and automatism. To approximate the roots of P ( x) , one can apply Gershgorin’s Theorem [5] or use matrix norms on a companion matrix to find disks in the complex plane to locate the eigenvalues [6]. From special companion matrices of a given polynomial P ( x) , improved methods of estimation of bounds for roots are developed by applying Gershgorin’s Theorem. The methodology of providing a sharp bound for polynomial’s roots consists of identifying a special form of Hessenberg matrix that could give an improved value of bound in case of submultiplicative matrix norms. The Sparse method is well appropriated to estimate bounds: it gives the opportunity to estimate improved bounds for high as well as minor values of coefficients for polynomial’s degree n = 5
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