1. Introduction
Let
be the real field. Consider an
m-th order
n dimensional tensor
which consists of
entries in
as follows.
A tensor is called nonnegative if Denote the set of all m-th order n-dimensional (nonnegative) tensors as (), and the set of all n-dimensional nonnegative (positive) vectors as ).
If there exists a complex number
and a nonzero complex vector
such that
then
is called an eigenvalue of
and
is termed as an eigenvetor of
associated with
and
are vectors whose
i-th entries are
and
, respectively, (see [
1]).
This definition was introduced by Qi [
2] for cases where
m is even and
is symmetric. Independently, this definition was also given by Lim [
3], where
and
were restricted to be a real vector and a real number, respectively. Similar to nonnegative matrix theory, the spectral radius of a tensor
is defined as
where
is the set of eigenvalues of a tensor
(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.
A nonnegative tensor is an important generalization of a nonnegative matrix, and there have been many results regarding the upper- and lower-bound estimations and numerical calculation of its spectral radius [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26]. Ng et al. [
11] proposed the NQZ algorithm (see Algorithm 1) and proved that the algorithm is applicable to the calculation of spectral radius of primitive tensors.
Algorithm 1:NQZ algorithm [11]. |
Step 0. Choose Let and set . Step 2. If , stop. Otherwise, replace k by and go to Step 1. |
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.
2. 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 is called reducible if there exists a nonempty proper index subset such thatIf is not reducible, then is irreducible.
Definition 2 ([
19]).
Let .(1) We call a nonnegative matrix the representation associated to nonnegative tensor , if the -th element of is defined to be the summation of with indices
(2) We call weakly reducible if its representation is a reducible matrix, and weakly primitive if is a primitive matrix. If is not weakly reducible, then it is called weakly irreducible.
Example 1. Let where and zero elsewhere, then Clearly, is irreducible and primitive, then tensor is weakly irreducible and weakly primitive. However, therefore, is reducible.
Theorem 1 ([
12]).
Let , is the spectral radius of , then Theorem 2 ([
16]).
For any nonnegative tensor , is an eigenvalue with a nonnegative eigenvector corresponding to it. Theorem 3 ([
26]).
If is weakly irreducible, then there exists a unique positive eigenvector of associated with . 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.
Definition 3 ([
1]).
Let , . The tensors and are said to be diagonal similar, if there exists some invertible diagonal matrix of order n such that , where . Theorem 4 ([
1]).
If the two m-th order n-dimensional tensors and are diagonal similar, then . Denote the set of
complex matrices as
. Let
. Denote the digragh of matrix
A as
(see [
4]).
Definition 4 ([
27]).
Let . is called strongly connected if for any pair of ordered nodes of , there is a directed path to connect them. Theorem 5 ([
27]).
If , then A is irreducible if and only if the digraph of A is strongly connected. Next, we will describe the algorithm for computing the spectral radius of weakly irreducible nonnegative tensors, which expands the existing results.
3. Algorithm
Let
,
Denote
,
. By Theorem 1, we have
For any
denote
then denote
. Use Theorem 1 and 4 to know that
Denote
then denote
similarly, denote
. By Theorem 1 and 4, we know
Denote
we obtain a sequence of weakly irreducible nonnegative tensors with the same zero-element pattern
and the sequence of positive vectors
.
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. |
Step 0. Given Compute
Set . Step 2. If , then , stop. Otherwise, set , go to Step 1. |
Because
is a positive diagonal matrix. Let
According to Theorem 7, Spec
Spec
For convenience we can take
, where
4. 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 is monotonically increasing and sequence is monotonically decreasing.
Lemma 1. Let , be the spectral radius of . For tensor sequence we have Proof. Since
, we have
where
Therefore
i.e.,
By Theorem 1 and Lemma 4, we have
□
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 if then Furthermore, for any holds true, where Proof. By PF algorithm, we know
Without loss of generality, assume
Because tensor
is weakly irreducible, there exists a directed path
from
to
by Definition 2, and
is on the digraph
of matrix
, where
such that
Therefore, combining Lemma 1, (
2) and
we have
i.e.,
Similarly, we get
So by (
3) and (
4), we obtain
that is,
By (
7) and (
5), we can get
that is,
Continue to proceed in turn, we obtain
i.e.,
Thus
Therefore, for any
we have
From Lemma 1, we have
□
Now we show our result on the convergence of the PF algorithm.
Theorem 6. If is weakly irreducible, is the spectral radius of , then .
Proof. we have
Similarly to the above formula, we can obtain
Since
is irreducible, for any
there exists a directed path
such that
. By Lemma 2, for any
, we have
Hence, we obtain
If
l is sufficiently large, we have
Together with (
8), for any
we have
Hence, we obtain
Thus, we have
Hence, we obtain
□
Remark 1. By Theorem 4, we know that Therefore, we know from the proof of Theorem 6 that PF algorithm is convergent for any weakly irreducible nonnegative tensors and any initial value .
5. 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 denote the order and the dimension of , respectively. denotes the spectral radius of . and denote the iteration of the PF algorithm and NQZ method, and denote the time costing by the PF algorithm and NQZ method, respectively.
Example 2. , where and zero elsewhere.
We know from Example 2, Clearly, is irreducible but not primitive, then tensor 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 , if , we obtain that the spectral radius of tensor is and the corresponding eigenvector is .
Remark 2. For the NQZ algorithm, the authors only prove the convergence of special weak primitive tensor , so the NQZ algorithm cannot be directly used to calculate the spectral radius of weak irreducible non-primitive nonnegative tensor.
Example 3. , where and zero elsewhere.
We know by Example 3,
Clearly,
is irreducible and primitive, then tensor
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
. The comparison results are presented in
Table 1, where
denotes error.
Example 4. Consider a random tensor , 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. Table 1 and
Table 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.
6. Conclusions
For general numerical algorithms in calculating the spectral radius of nonnegative tensors , 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, or transformed , where 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.