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.


Introduction
Let R be the real field.Consider an m-th order n dimensional tensor A which consists of n m entries in R as follows.
A = (a i 1 i 2 •••i m ) n i 1 ,i 2 ,...,i m =1 , where Denote the set of all m-th order n-dimensional (nonnegative) tensors as R [m,n] (R [m,n] + ), and the set of all n-dimensional nonnegative (positive) vectors as R n + (R n ++ ).If there exists a complex number λ and a nonzero complex vector x = (x 1 , x 2 , . . ., x n ) T such that then λ is called an eigenvalue of A and x is termed as an eigenvetor of A associated with λ, Ax m−1 and x [m−1] are vectors whose i-th entries are , respectively, (see [1]).This definition was introduced by Qi [2] for cases where m is even and A is symmetric.Independently, this definition was also given by Lim [3], where x and λ were restricted to be a real vector and a real number, respectively.Similar to nonnegative matrix theory, the spectral radius of a tensor A is defined as ρ(A) = sup{| λ | : λ ∈ spec(A)}, where spec(A) is the set of eigenvalues of a tensor A(see [4]).
In recent years, the eigenvalue problems of nonnegative tensors have arisen in a wide range of practical applications such as algebraic geometry [5], spectral hypergraph theory [6,7], higher-order Markov chain [8,9] and so on.In particular, the spectral theory of tensors remains an important research topic in this field.
Chang et al. [14] proved the convergence of the NQZ method for primitive tensors.Zhang and Qi [21] gave the linear convergence rate of the NQZ method for essentially positive tensors.Hu et al. [16] provided the R-linear convergence rate of the NQZ method for weakly primitive tensors, and proposed an algorithm for finding the spectral radius of weakly irreducible nonnegative tensors.In order to hasten the convergence of the NQZ method, Yang et al. [19] proposed a method with parameters, and proved that the algorithm has an explicit linear convergence rate for indirectly positive tensors, and the algorithm coincides with the NQZ method assuming the parameters are properly selected.
In order to render the NQZ algorithm applicable to the calculation of spectral radius of more types of nonnegative tensors, many scholars have performed an abundance of research.For example, in the literature [23], the authors first transform the general nonnegative tensor into the weakly irreducible form of the tensor, and then use the NQZ method to calculate the spectral radius of the weakly irreducible block successively.Similar to the literature [23], the authors in [22] transform a general symmetric nonnegative tensor into a weakly irreducible form of the tensor and calculate its spectral radius.In [24], an algorithm for calculating the spectral radius of a class of irreducible nonnegative tensors and generalized weakly irreducible nonnegative tensors (Definition 4.2 of [24]) is proposed.In [25], an inverse iterative method with local quadratic convergence for calculating the spectral radius of weakly irreducible nonnegative tensors is given, in which an LU decomposition of the matrix is needed in each iteration.
The above methods, except for the research work [25], all stem from the ideas of the NQZ method.Different from the NQZ method, we will apply the diagonal similarity transformation of tensors to construct an algorithm for computing the spectral radius of a weakly irreducible tensor with the aid of a power function.We also prove that this algorithm is convergent for all weakly irreducible nonnegative tensors.

Preliminaries
In this section, we mainly introduce some related concepts and important properties of tensors and matrices.

Definition 1 ([10]
).An m-th order n-dimensional tensor A is called reducible if there exists a nonempty proper index subset I ⊂ n such that (1) We call a nonnegative matrix G(A) the representation associated to nonnegative tensor A, if the (i, j)-th element of G(A) is defined to be the summation of (2) We call A weakly reducible if its representation G(A) is a reducible matrix, and weakly primitive if G(A) is a primitive matrix.If A is not weakly reducible, then it is called weakly irreducible.A is weakly irreducible and weakly primitive.However, a 2i 2 i 3 = 0, i 2 , i 3 ∈ {1, 3}, therefore, A is reducible.

Theorem 2 ([16]). For any nonnegative tensor
is weakly irreducible, then there exists a unique positive eigenvector of A associated with ρ(A).
Next, we state a special kind of similarity of tensors: the diagonal similarity.This similarity relation among tensors plays an important role in the study of the spectra of nonnegative irreducible tensors. ,n] .The tensors A and B are said to be diagonal similar, if there exists some invertible diagonal matrix D

Theorem 4 ([1]
).If the two m-th order n-dimensional tensors A and B are diagonal similar, then spec(A) = spec(B).
Denote the set of n × n complex matrices as C n×n .Let A = (a ij ) ∈ C n×n .Denote the digragh of matrix A as Γ(A)(see [4]).
Next, we will describe the algorithm for computing the spectral radius of weakly irreducible nonnegative tensors, which expands the existing results.
From the construction process of the above nonnegative tensor sequences, we propose the following algorithm for finding the spectral radius of weakly irreducible nonnegative tensors.We call the algorithm the power function (PF) algorithm (see Algorithm 2).Algorithm 2: PF algorithm.

Convergence of the PF Algorithm
In this section, we prove that PF algorithm is convergent for weakly irreducible nonnegative tensors.
Firstly, we show that the sequence {λ + , ρ(A) be the spectral radius of A. For tensor sequence A (l) (l = 0, 1, 2, • • • ), we have Similarly, By Theorem 1 and Lemma 4, we have Hence, we obtain Before proving the convergence of PF algorithm, we first prove that the entries of the tensor sequence generated by PF algorithm have a lower bound.

Lemma 2. Let
be weakly irreducible.For tensor sequence A (l) = (a (l) holds true, where Proof.By PF algorithm, we know ( Because tensor A is weakly irreducible, there exists a directed path γ from t 1 to t n by Definition 2, and γ is on the digraph Γ(G(A)) of matrix G(A), where γ : Therefore, combining Lemma 1, (2) and l ∈ Z + ∪ {0}, we have Similarly, we get . . .
Continue to proceed in turn, we obtain Therefore, for any From Lemma 1, we have 0 < a λ ≤ 1.
Now we show our result on the convergence of the PF algorithm.
Similarly to the above formula, we can obtain Since A is irreducible, for any i ∈ {t s ∈ n : λ (0) Hence, we obtain If l is sufficiently large, we have . Together with (8), for any l ∈ Z + ∪ {0}, we have Hence, we obtain Thus, we have Hence, we obtain lim Remark 1.By Theorem 4, we know that spec(A) = spec(A (0) ) = spec(A (1) ).Therefore, we know from the proof of Theorem 6 that PF algorithm is convergent for any weakly irreducible nonnegative tensors and any initial value x (0) ∈ R n ++ .

Numerical Examples
In this section, we provide some numerical results and comparison for convergence rate of calculating the spectral radius of weakly irreducible nonnegative between the PF algorithm and NQZ method [11].All numerical experiments are conducted using Matlab R2016b on a PC with 4GB memory Intel CPU 15-4210.
The numbers m and n in (m, n) denote the order and the dimension of A, respectively.ρ(A) denotes the spectral radius of A. iter PF and iter NQZ denote the iteration of the PF algorithm and NQZ method, CPU PF (s) and CPU NQZ (s) denote the time costing by the PF algorithm and NQZ method, respectively.Example 2. A = (a ijk ) ∈ R [3,3] + , where a 122 = 2, a 233 = 3, a 311 = 4, and zero elsewhere.
We know from Example 2, G(A) =   0 4 0 0 0 6 8 0 0   .Clearly, G(A) is irreducible but not primitive, then tensor A is weakly irreducible but not weakly primitive.However, the condition of convergence of PF algorithm is satisfied, therefore, we can use the PF algorithm to calculate the spectral radius and corresponding eigenvector of the tensor.Take α = 1 2 , if ε = 10 −5 , we obtain that the spectral radius of tensor A is ρ(A) = 2.88450, and the corresponding eigenvector is x = (0.75161, 0.90263, 0.88509) T .Remark 2. For the NQZ algorithm, the authors only prove the convergence of special weak primitive tensor A = (a , so the NQZ algorithm cannot be directly used to calculate the spectral radius of weak irreducible non-primitive nonnegative tensor.

We know by Example 3, G(
Clearly, G(A) is irreducible and primitive, then tensor A is weakly irreducible and weakly primitive.Therefore, the PF algorithm and NQZ method can be used to calculate the spectral radius and corresponding eigenvector of the tensor.Take α = 1 2 .The comparison results are presented in Table 1, where ε denotes error.+ , whose all entries are random values drawn from the standard uniform distribution on (0, 1).Obviously, this is a nonnegative irreducible tensor.The comparison results are presented in Table 2, where ε denotes error.Tables 1 and 2 show a comparison between the PF algorithm and the NQZ method given in [11], with the same error, the number of iterations and calculation time are significantly reduced, which further verifies that our proposed algorithm is more efficient.

Conclusions
For general numerical algorithms in calculating the spectral radius of nonnegative tensors A = (a + , the research results regarding the NQZ algorithm are the richest.At present, however, it is only proved that the NQZ algorithm is applicable to the calculation of spectral radius of special weak primitive tensors, that is, a ii•••i > 0, i = 1, 2, • • • , n or transformed A + αI, α > 0, where A is a weakly irreducible weak primitive tensor.In this paper, a numerical algorithm for calculating the spectral radius of nonnegative tensors, with the aid of power functions, is constructed by using the diagonal similarity transformation of tensors.The algorithm is applicable to the calculation of spectral radius of all weakly irreducible nonnegative tensors.Moreover, the algorithm not only bears a wider scope of application, but also has a higher computational efficiency compared with the NQZ method.

Table 1 .
The comparison of the PF algorithm and NQZ method in Example 3.
Example 4. Consider a random tensor A ∈ R [m,n]

Table 2 .
The comparison of PF algorithm and NQZ method in Example 4.