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).


Introduction
Let C (R) be the set of all complex (real) numbers, R + (R ++ ) be the set of all nonnegative (positive) numbers, C n (R n ) be the set of all dimension n complex (real) vectors, and R n + (R n ++ ) be the set of all dimension n nonnegative (positive) vectors. An order m dimension n complex (real) tensor A = (a i  i  ···i m ), denoted by A ∈ C [m,n] (A ∈ R [m,n] , respectively), consists of n m entries: a i  i  ···i m ∈ C (R), ∀i k ∈ N = {, , . . . , n}, k = , , . . . , m.
As the eigenvalues of matrices have many extensive applications, the H-eigenvalues [] for higher order tensors also have a wide range of applications such as numerical multilinear algebra and higher order Markov chains [-]. [m,n] . Then (λ, x) ∈ C × C n \{} is called an eigenpair of A if

Definition  ([]) Let
If A is not reducible, then we call A irreducible.
A is called weakly irreducible if M is an irreducible matrix.

n] be a nonnegative tensor, then ρ(A) ≥  is an eigenvalue of A with a nonnegative eigenvector x corresponding to it.
For the spectral radius of a nonnegative tensor A, although some algorithms of calculating its value were proposed [-], it is not easy to choose an appropriate iterative initial value such that these iterative methods rapidly converge to its exact value. Therefore, it is necessary to give an initial estimate for the spectral radius of a nonnegative tensor. Actually, there are already some results for the bound of the nonnegative tensors' spectral radius, for example, Yang and Yang extended the classical spectral radius bound for nonnegative matrices to nonnegative tensors in [] and obtained the following result.
In [], by estimating the ratio of the smallest component and the largest component of a Perron vector, Li and Ng gave the following bound for the spectral radius of a nonnegative tensor and proved it is better than the bound in (). where In this paper, we continue to study this problem and present a new lower bound and a new upper bound for the spectral radius of a nonnegative tensor by giving a new ratio of the smallest component and the largest component of a Perron vector. It is proved that this bound is better than the bound in (). Numerical examples are also given to illustrate the efficiency of the proposed results.

Bounds for the spectral radius of nonnegative tensors
In this section, we first give a lemma to estimate the ratio of the smallest component and the largest component of a Perron vector, and then we give a bound for the spectral radius of nonnegative tensors.
Proof Since A is a weakly irreducible nonnegative tensor, according to Theorem , we Similarly, we have that for each i ∈ N , Taking i = q in (), we have that q on both sides of () gives Combining () with () gives Multiplying ( x s x l ) m- on both sides of () gives Note that it is not easy to get the bound of x s x l simply from (); however, we can overcome this difficulty by using the fact that  ≤ x s x l ≤  for the right-hand side of (). Hence by () we have that , which together with () yields The conclusion follows.

n] be a weakly irreducible nonnegative tensor. Then
where Combining () with Lemma  gives Similarly, by the first inequality of (), we have that for each i ∈ N , The proof is completed.
Remark  It is easy to see that the bound in () also holds for general nonnegative tensors. In fact, if A = (a i  i  ···i m ) ∈ R [m,n] is a nonnegative tensor, and F = ( with f i  i  ···i m =  for all i r ∈ N , r = , , . . . , m, then A + εF is a weakly irreducible tensor for any ε > . Hence by Theorem  we can give the bound of ρ(A + εF ). Since the spectral radius of a nonnegative tensor is a continuous function of its entries, the bound for ρ(A) can be obtained when ε → , which is exactly the bound in ().
Remark  Note that the first inequality of () can be replaced by then, similar to the proof of Lemma , we can obtain that where And hence, by the similar proof of Theorem , we can give another bound of spectral radius for a nonnegative weakly irreducible tensor A as follows. Although the bound in () is not better than the bound in (), it needs less computations.

Corollary  Let
Next is a comparison result for the bound in () and the bound in ().

Results and discussion
The main result of this paper is Theorem . From Remark  and the proof of Lemma , it is not difficult to see that the right expressions of last inequality () can also be replaced by many similar expressions according to the extent of magnifying inequality. Therefore, we can also obtain other bounds for the spectral radius of a nonnegative tensor. Furthermore, we notice that the bound in () is the best of them as the last inequality () reaches the optimum for all those possible expressions, which can be shown by two numerical examples above.

Conclusions
In this paper, we propose a new bound for the spectral radius of a nonnegative tensor by estimating the ratio of the smallest component and the largest component of a Perron vector. And we prove that the proposed result improves the bound in [].