Author(s)

Mamoudou Amadou Bondabou¹, Ousmane Moussa Tessa¹, Amidou Morou²

¹Département de Mathématiques et d’Informatique, Université Abdou Moumouni, Niamey, Niger.
²Institut de Recherche sur l’Enseignement des Mathématiques, Université A. Moumouni de Niamey, Niamey, Niger.

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.

Fiedler Matrices, Polynomial’s Roots, Bounds for Polynomials, Companion Matrices, Sparse Companion Matrices, Hessenberg Matrices, Submultiplicative Matrix Norm

Bondabou, M. , Tessa, O. and Morou, A. (2021) Bounds for Polynomial’s Roots from Fiedler and Sparse Companion Matrices for Submultiplicative Matrix Norms. Advances in Linear Algebra & Matrix Theory, 11, 1-13. doi: 10.4236/alamt.2021.111001.