Perron vector analysis for irreducible nonnegative tensors and its applications

Wen Li; Wei-Hui Liu; Seak Weng Vong

doi:10.3934/jimo.2019097

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2021, Volume 17, Issue 1: 29-50. Doi: 10.3934/jimo.2019097

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Perron vector analysis for irreducible nonnegative tensors and its applications

1.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2.
Department of Mathematics, University of Macau, Macau, China

^* Corresponding author: Wei-Hui Liu
^* Corresponding author: Wei-Hui Liu

Received: December 31, 2018

Revised: February 28, 2019

Early access: July 2019

Published: January 2021

The first author is supported by NSFC grant 11671158, U1811464 and 11771159. The second author is supported by NSFC grant 11571124 and UM grant MYRG2016-00077-FST. The third author is supported by UM grant MYRG2017-00098-FST

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Abstract

In this paper, we analyse the Perron vector of an irreducible nonnegative tensor, and present some lower and upper bounds for the ratio of the smallest and largest entries of a Perron vector based on some new techniques, which always improve the existing ones. Applying these new ratio results, we first refine two-sided bounds for the spectral radius of an irreducible nonnegative tensor. In particular, for the matrix case, the new bounds also improve the corresponding ones. Second, we provide a new Ky Fan type theorem, which improves the existing one. Third, we refine the perturbation bound for the spectral radii of nonnegative tensors, from which one may derive a comparison theorem for spectral radii of nonnegative tensors. Numerical examples are given to show the efficiency of the theoretical results.

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Mathematics Subject Classification: Primary: 15A69.

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Figure 1. Comparison between two Ky Fan type Theorems

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Figure 2. The results of randomly constructed tensors

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Figure 3. The results of perturbation bounds (left) n = 5 and (right) n = 10

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Table 1. Comparisons with the upper bounds for the ratio

$ \omega_1 $ in (3)	$ \omega_2 $ in (4)	$ \omega_3 $ in (5)	$ \omega_4 $ in (11)
0.5575	0.5307	0.4855	0.5244

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Table 2. Comparisons with the lower bounds for ratio

	Example 2	Example 3	Example 4
Actual value of $ \frac{x_{\min}}{x_{\max}} $	0.9873	0.6402	0.6794
$ \kappa_0 $ in (2)	0.6300	0.3969	0.5000
$ \kappa_1 $ in (3)	0.7857	0.2083	0.3077
$ \kappa_2 $ in (4)	0.5848	0.5000	0.4642
$ \kappa_3^{(1)} $ in (13)	$ {\bf{0.9662}} $	0.2808	0.3445
$ \kappa_3^{(1)} $ in (13)	(t = -5.5602)	(t = -5.7276)	(t = -5.0250)
$ \kappa_3^{(2)} $ in (16)	0.9258	0.5000	$ {\bf{0.5539}} $
(also in (13))	(t = -5.1168)	(t = -2.2315)	(t = -2.2956)
$ \kappa_3^{(3)} $ in (13)	0.6300	$ {\bf{0.5724}} $	0.5503
$ \kappa_3^{(3)} $ in (13)	(t = -3.0000)	(t = -3.5887)	(t = -2.5208)
$ \kappa_3 $ in (13)	$ {\bf{0.9662}} $	$ {\bf{0.5724}} $	$ {\bf{0.5539}} $

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Table 3. Comparisons between (20) and (27)

Dimension	$ n = 5 $	$ n = 10 $	$ n = 15 $	$ n = 20 $
Lower bound	42.86%	64.02%	75.64%	81.37%
Upper bound	91.76%	94.50%	95.77%	96.83%

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