The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis

Zhongming Chen, Liqun Qi, Qingzhi Yang, Yuning Yang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

11 Citations (Scopus)

Abstract

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.
Original languageEnglish
Pages (from-to)3713-3733
Number of pages21
JournalLinear Algebra and Its Applications
Volume439
Issue number12
DOIs
Publication statusPublished - 15 Dec 2013

Keywords

  • Algorithm
  • Complexity
  • Convergence
  • Eigenvalue
  • Nonnegative tensor
  • Perturbation
  • Singular value

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Numerical Analysis

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