The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor

Abstract. In this paper, we study the perturbation bound for the spectral radius of an mthorder n-dimensional non-negative tensor A. The main contribution of this paper is to show that when A is perturbed to a non-negative tensor à by ∆A, the absolute difference between the spectral radii of A and à is bounded by the largest magnitude of the ratio of the i-th component of ∆Axm−1 and the i-th component xm−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 non-negative tensor in general. On the other hand, we study the backward error matrix ∆A and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible non-negative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible non-negative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.