The point-wise convergence of shifted symmetric higher order power method

Gang Luo; Qingzhi Yang

doi:10.3934/jimo.2019115

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2021, Volume 17, Issue 1: 357-368. Doi: 10.3934/jimo.2019115

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The point-wise convergence of shifted symmetric higher order power method

Gang Luo^1, and
Qingzhi Yang^1,2, ,

1.
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
2.
School of Mathematics and Statistics, Kashi University, Kashi 844006, China

^* Corresponding author: Qingzhi Yang
^* Corresponding author: Qingzhi Yang

Received: December 31, 2018

Revised: May 31, 2019

Early access: September 2019

Published: January 2021

The second author is supported by Natural Science Foundation of Xinjiang (Grant No. 2017D01A14)

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Abstract

Shifted symmetric higher-order power method (SS-HOPM) is an effective method of computing tensor eigenpairs. However the point-wise convergence of SS-HOPM has not been proven yet. In this paper, we provide a solid proof of the point-wise convergence of SS-HOPM via Łojasiewicz inequality. In particular, we establish a mapping from the sequence generated by the algorithm to a specially defined sequence. Using Łojasiewicz inequality, we prove the convergence of the new sequence, then the original sequence is convergent based on the relation of two sequences.

Keywords:

Z-eigenvalue,
shifted symmetric higher-order power method,
point-wise convergence,
Łojasiewicz inequality.

Mathematics Subject Classification: Primary: 15A18, 90C26; Secondary: 15A69.

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Figure 1. Trajectories of sequences $ \{x_k\} $, $ \{y_k\} $ generated by SS-HOPM

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