Bounds for Polynomial’s Roots from Hessenberg Matrices and Gershgorin’s Disks  

作　　者：Mamoudou Amadou Bondabou Ousmane Moussa Tessa Maimouna Salou Mamoudou Amadou Bondabou;Ousmane Moussa Tessa;Maimouna Salou(Département de Mathématiques et d’informatique, Université A. Moumouni, Niamey, Niger)

机构地区：[1]Département de Mathématiques et d’informatique, Université A. Moumouni, Niamey, Niger

出　　处：《Advances in Pure Mathematics》2021年第12期963-977,共15页理论数学进展（英文）

摘　　要：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.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 

分 类 号：O17[理学—数学]