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Perron vector analysis for irreducible nonnegative tensors and its applications

  • * Corresponding author: Wei-Hui Liu

    * Corresponding author: Wei-Hui Liu 

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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  • 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.

    Mathematics Subject Classification: Primary: 15A69.

    Citation:

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  • Figure 1.  Comparison between two Ky Fan type Theorems

    Figure 2.  The results of randomly constructed tensors

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

    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
     | Show Table
    DownLoad: CSV

    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
    (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
    (t = -3.0000) (t = -3.5887) (t = -2.5208)
    $ \kappa_3 $ in (13) $ {\bf{0.9662}} $ $ {\bf{0.5724}} $ $ {\bf{0.5539}} $
     | Show Table
    DownLoad: CSV

    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%
     | Show Table
    DownLoad: CSV
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