The Equivalent Forms Of The Ostrowski Upper Bounds And New Bounds For The Spectral Radius Of Nonnegative Matrices

A.M. Ostrowski in1951gave two well-known upper bounds for the spectral radius of nonnegative matrices. However, the bounds are not of much practical use because they all involve a parameter a in the interval [0,1], and it is not easy to determine the best value of α€[0,1] in application. In this thesis, we study this problem and equivalent forms of Ostrowski’s upper bounds are given which can be computed with the entries of matrix and without having to minimize the expressions of the bounds over all possible values of α∈[0,1]. In addition, a new upper bound of the spectral radius of nonnegative matrices is given by applying the localization theorem of eigenvalues, and prove that the upper bound is less than or equal to Brauer-Gentry’s. The numerical examples show that the new upper bound is better than Brauer-Gentry’s bound and A. Melman’s bound which is obtained recently [A.Melman.Upper and lower bounds for the Perron root of a nonnegative matrix, Linear and Multilinear Algebra,2013,61(2):171-181].