Noda Iteration for Computing Generalized Tensor Eigenpairs

03/02/2023

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by   Wanli Ma, et al.

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In this paper, we propose the tensor Noda iteration (NI) and its inexact version for solving the eigenvalue problem of a particular class of tensor pairs called generalized ℳ-tensor pairs. A generalized ℳ-tensor pair consists of a weakly irreducible nonnegative tensor and a nonsingular ℳ-tensor within a linear combination. It is shown that any generalized ℳ-tensor pair admits a unique positive generalized eigenvalue with a positive eigenvector. A modified tensor Noda iteration(MTNI) is developed for extending the Noda iteration for nonnegative matrix eigenproblems. In addition, the inexact generalized tensor Noda iteration method (IGTNI) and the generalized Newton-Noda iteration method (GNNI) are also introduced for more efficient implementations and faster convergence. Under a mild assumption on the initial values, the convergence of these algorithms is guaranteed. The efficiency of these algorithms is illustrated by numerical experiments.