Strictly nonnegative tensors and nonnegative tensor partition |
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Authors: | ShengLong Hu ZhengHai Huang LiQun Qi |
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Institution: | 1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China 2. Department of Mathematics, School of Science, Tianjin University, Tianjin, 300072, China 3. The Center for Applied Mathematics of Tianjin University, Tianjin University, Tianjin, 300072, China
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Abstract: | 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 necessary and sufficient 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. Some preliminary numerical results show the feasibility and effectiveness of the algorithm. |
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Keywords: | nonnegative tensor spectral radius strict nonnegativity weak irreducibility |
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