Strictly nonnegative tensors and nonnegative tensor partition

• DOI: 10.1007/s11425-013-4752-4
• Publication Date: 2013-11-29T07:43:07Z
• Authors: ShengLong Hu; ZhengHai Huang; LiQun Qi
• Keywords: 15-02, 15A18, 15A69, 65F15; FOS: Mathematics; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); 0101 mathematics; 01 natural sciences

In this paper, we introduce a new class of nonnegative tensors --- strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T, there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.<br/>25 pages, 1 figue<br/>