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Bounds for Polynomial’s Roots from hessenberg matrices and Gershgorin’s Disks

Advances in Pure Mathematics 2021年第12期11卷 963-977页

作者： Mamoudou Amadou Bondabou Ousmane Moussa Tessa Maimouna Salou Département de Mathématiques et d’informatique Université A. Moumouni Niamey Niger

The goal of this study is to propose a method of estimation of bounds for roots of polynomials with complex coefficients. A well-known and easy tool to obtain such information is to use the standard Gershgorin’s theorem, however, it doesn’t take into account the structure of the matrix. The modified disks of Gershgorin give the opportunity through some geometrical figures called Ovals of Cassini, to consider the form of the matrix in order to determine appropriated bounds for roots. Furthermore, we have seen that, the Hessenbeg matrices are indicated to estimate good bounds for roots of polynomials as far as we become improved bounds for high values of polynomial’s coefficients. But the bounds are better for small values. The aim of the work was to take advantages of this, after introducing the Dehmer’s bound, to find an appropriated property of the hessenberg form. To illustrate our results, illustrative examples are given to compare the obtained bounds to those obtained through classical methods like Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds.

关键词： Bounds for Roots of Polynomials Gershgorin Frobenius Companion Matrix hessenberg matrices Ovals of Cassini

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

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Advances in Linear Algebra & Matrix Theory 2021年第1期11卷 1-13页

作者： Mamoudou Amadou Bondabou Ousmane Moussa Tessa Amidou Morou Département de Mathématiques et d’Informatique Université Abdou Moumouni Niamey Niger Institut 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 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.

关键词： Fiedler matrices Polynomial’s Roots Bounds for Polynomials Companion matrices Sparse Companion matrices hessenberg matrices Submultiplicative Matrix Norm

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