A power-like method for finding the spectral radius of a weakly irreducible nonnegative symmetric tensor

Xueli Bai (), Dong-Hui Li (), Lei Wu () and Jiefeng Xu ()
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Xueli Bai: Guangdong University of Foreign Studies
Dong-Hui Li: South China Normal University
Lei Wu: Jiangxi Normal University
Jiefeng Xu: South China Normal University

Computational Optimization and Applications, 2024, vol. 89, issue 3, No 10, 895-926

Abstract: Abstract The Perron–Frobenius theorem says that the spectral radius of a weakly irreducible nonnegative tensor is the unique positive eigenvalue corresponding to a positive eigenvector. With this fact in mind, the purpose of this paper is to find the spectral radius and its corresponding positive eigenvector of a weakly irreducible nonnegative symmetric tensor. By transforming the eigenvalue problem into an equivalent problem of minimizing a concave function on a closed convex set, we derive a simpler and cheaper iterative method called power-like method, which is well-defined. Furthermore, we show that both sequences of the eigenvalue estimates and the eigenvector evaluations generated by the power-like method Q-linearly converge to the spectral radius and its corresponding eigenvector, respectively. To accelerate the method, we introduce a line search technique. The improved method retains the same convergence property as the original version. Plentiful numerical results show that the improved method performs quite well.

Keywords: Weakly irreducible nonnegative tensor; Spectral radius; Power method; Q-linear convergence; 15A18; 15A69; 90C90 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-024-00601-8

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