$ \omega_1 $ in (3) | $ \omega_2 $ in (4) | $ \omega_3 $ in (5) | $ \omega_4 $ in (11) |
0.5575 | 0.5307 | 0.4855 | 0.5244 |
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.
Citation: |
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 |
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}} $ |
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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Comparison between two Ky Fan type Theorems
The results of randomly constructed tensors
The results of perturbation bounds (left) n = 5 and (right) n = 10