Spectral Radius Of Nonnegative Tensors Research Articles

Quasi-Double Diagonally Dominant H-Tensors and the Estimation Inequalities for the Spectral Radius of Nonnegative Tensors

In this paper, we study two classes of quasi-double diagonally dominant tensors and prove they are H-tensors. Numerical examples show that two classes of H-tensors are mutually exclusive. Thus, we extend the decision conditions of H-tensors. Based on these two classes of tensors, two estimation inequalities for the upper and lower bounds for the spectral radius of nonnegative tensors are obtained.

Bounds for spectral radius of nonnegative tensors using matrix-digragh-based approach

<p style='text-indent:20px;'>We obtain the improved results of the upper and lower bounds for the spectral radius of a nonnegative tensor by its majorization matrix's digraph. Numerical examples are also given to show that our results are significantly superior to the results of related literature.</p>

Power Function Method for Finding the Spectral Radius of Weakly Irreducible Nonnegative Tensors

Since the eigenvalue problem of real supersymmetric tensors was proposed, there have been many results regarding the numerical algorithms for computing the spectral radius of nonnegative tensors, among which the NQZ method is the most studied. However, the NQZ method is only suitable for calculating the spectral radius of a special weakly primitive tensor, or a weakly irreducible primitive tensor that satisfies certain conditions. In this paper, by means of diagonal similarrity transformation of tensors, we construct a numerical algorithm for computing the spectral radius of nonnegative tensors with the aid of power functions. This algorithm is suitable for the calculation of the spectral radius of all weakly irreducible nonnegative tensors. Furthermore, it is efficient and can be widely applied.

Some inequalities on the spectral radius of nonnegative tensors

Abstract The eigenvalues and the spectral radius of nonnegative tensors have been extensively studied in recent years. In this paper, we investigate the analytic properties of nonnegative tensors and give some inequalities on the spectral radius.

The bounds on spectral radius of nonnegative tensors via general product of tensors

In [1], J.Y. Shao proposed the general product of tensors. Based on this product, the Minc-type bound on spectral radius of nonnegative tensors is given. And the generalized Minimax Theorem of nonnegative tensors is presented. Moreover, we give some bounds on the spectral radius of the general product of tensors.

An algorithm for computing the spectral radius of nonnegative tensors

In this paper, we present an algorithm to find the weakly irreducible normal form of tensors. Based on the weakly irreducible normal form of nonnegative tensors, we present a convergent algorithm for computing the spectral radius of any nonnegative tensors. Numerical results are reported to show that the proposed algorithm is efficient and also able to compute the spectral radius of any nonnegative tensors.

A new estimate for the spectral radius of nonnegative tensors

In this paper, we are concerned with the spectral radius of nonnegative tensors. By estimating the ratio of the smallest component and the largest component of a Perron vector, a new bound for the spectral radius of nonnegative tensors is obtained. It is proved that the new bound improves some existing ones. Finally, a numerical example is implemented to show the effectiveness of the proposed bound.

A new bound for the spectral radius of nonnegative tensors

By estimating the ratio of the smallest component and the largest component of a Perron vector, we provide a new bound for the spectral radius of a nonnegative tensor. And it is proved that the proposed result improves the bound in (Li and Ng in Numer. Math. 130(2):315-335, 2015).

A new S-type eigenvalue inclusion set for tensors and its applications.

In this paper, a new S-type eigenvalue localization set for a tensor is derived by dividing N={1,2,ldots,n} into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of strong M-tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).

A Method with Parameter for Solving the Spectral Radius of Nonnegative Tensor

In this paper, a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, we prove this method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method (proposed by Ng, Qi and Zhou in Finding the largest eigenvalue of a non-negative tensor SIAM J Matrix Anal Appl 31:1090–1099, 2009) by taking a specific parameter. Furthermore, we give a modified NQZ method, which has an explicit linear convergence rate for nonnegative tensors and has an error bound for nonnegative tensors with a positive Perron vector. Besides, we promote an inexact power-type algorithm. Finally, some numerical results are reported.

A new Brauer-type eigenvalue localization set for tensors

A new Brauer-type eigenvalue localization set for tensors is given. We show that this set is tighter than those provided by Qi (J. Symbolic Comput. 2005 ), Li et al. (Numer. Linear Algebra App. 2014), and Li and Li (Linear and Multilinear Algebra, 2015). Based on this set, we give an upper bound for the spectral radius of nonnegative tensors, and prove that the new bound is sharper than those provided by Yang and Yang (SIAM J. Matrix Anal. App. 2010) and Li et al. (Numer. Linear Algebra App. 2014).

Bounds for the spectral radius of nonnegative tensors

Lower bounds and upper bounds for the spectral radius of a nonnegative tensor are provided. And it is proved that these bounds are better than the corresponding bounds in [Y. Yang, Q. Yang, Further results for Perron-Frobenius Theorem for nonnegative tensors, SIAM. J. Matrix Anal. Appl. 31 (2010), 2517-2530].

A new eigenvalue inclusion set for tensors and its applications

A new tensor eigenvalue inclusion set is given, and proved to be tighter than those in L.Q. Qi (2005) [18] and C.Q. Li, Y.T. Li, X. Kong (2014) [12]. In addition, we study the eigenvalues lying on the boundary of the eigenvalue inclusion set provided by Qi (2005) for weakly irreducible tensors. As applications, we give new bounds for the spectral radius of nonnegative tensors, some sufficient conditions for a tensor to be a strong M-tensor and some inequalities to identify the positive definiteness for an even-order real supersymmetric tensor.

The Perturbation Bound for the Spectral Radius of a Nonnegative Tensor

We study the perturbation bound for the spectral radius of an mth-order n-dimensional nonnegative tensor A. The main contribution of this paper is to show that when A is perturbed to a nonnegative tensor A~ by ΔA, the absolute difference between the spectral radii of A and A~ is bounded by the largest magnitude of the ratio of the ith component of ΔAxm-1 and the ith component xm-1, where x is an eigenvector associated with the largest eigenvalue of A in magnitude and its entries are positive. We further derive the bound in terms of the entries of A only when x is not known in advance. Based on the perturbation analysis, we make use of the NQZ algorithm to estimate the spectral radius of a nonnegative tensor in general. On the other hand, we study the backward error matrix ΔA and obtain its smallest error bound for its perturbed largest eigenvalue and associated eigenvector of an irreducible nonnegative tensor. Based on the backward error analysis, we can estimate the stability of computation of the largest eigenvalue of an irreducible nonnegative tensor by the NQZ algorithm. Numerical examples are presented to illustrate the theoretical results of our perturbation analysis.

Some bounds for the spectral radius of nonnegative tensors

In this paper, we extend the well-known column sum bound of the spectral radius for nonnegative matrices to the tensor case, an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor. Also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors. We demonstrate that our new results improve existing results. The other main results of this paper is to provide a sharper Ky Fan type theorem and a comparison theorem for nonnegative tensors. Finally, we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors. This result is similar to the Weyl theorem for the matrix case.

Strictly nonnegative tensors and nonnegative tensor partition

In this paper, we introduce a new class of nonnegative tensors --- strictly nonnegative tensors. A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa. We show that the spectral radius of a strictly nonnegative tensor is always positive. We give some sufficient and necessary conditions for the six well-conditional classes of nonnegative tensors, introduced in the literature, and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors. We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility. We show that for a nonnegative tensor T, there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible; and the spectral radius of T can be obtained from those spectral radii of the induced tensors. In this way, we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption. The preliminary numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.

Further Results for Perron–Frobenius Theorem for Nonnegative Tensors

For a nonnegative irreducible tensor, we give distribution properties of its eigenvalues. In particular, the spectral radius of a nonnegative irreducible tensor with positive trace is proved to be the unique eigenvalue on the spectral circle. Unlike the matrix setting, we give an example to present that this type of tensor is not always primitive. Thus, for a nonnegative irreducible tensor, the primitivity is a sufficient condition only for the spectral radius to be the unique eigenvalue on the spectral circle. Also, the stochastic tensor is defined, and we show that every nonnegative irreducible tensor with unit spectral radius is diagonally similar to a certain irreducible stochastic tensor. Based on this result, the minimax theorem for tensors is proved by using an alternative approach. Further, with the help of the minimax theorem, we illustrate that the problem of finding the spectral radius (largest singular value) of a nonnegative irreducible square (rectangular) tensor can be converted into a convex optimization problem. Additionally, we give an equivalent condition of irreducible nonnegative tensors. By this condition, one can easily determine whether or not a nonnegative tensor is irreducible.