The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor. (28th September 2014)

• Record Type: Journal Article
• Authors: Li, Wen; Ng, Michael K.
• Other Names: Risebro Nils Henrik Academic Editor.
• Is Part Of: Advances in numerical analysis. Volume 2014(2014)
• Publication Date: 2014-09-28
• Subjects: Numerical analysis -- Periodicals; Numerical analysis; Periodicals; Electronic journals; 518
• DOI: 10.1155/2014/109525
• Languages: English
• ISSNs: 1687-9562

Abstract

Abstract : We study the perturbation bound for the spectral radius of an m th-order n -dimensional nonnegative tensor A . The main contribution of this paper is to show that when A is perturbed to a nonnegative tensor A ~ by Δ A, the absolute difference between the spectral radii of A and A ~ is bounded by the largest magnitude of the ratio of the i th component of Δ A x m - 1 and the i th component x m - 1, where x is an eigenvector associated with the largest eigenvalue of A in magnitude and its entries are positive. We further derive the bound in terms of the entries of A only when x is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix Δ A and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.