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.


Introduction
Eigenvalue problems of higher order tensors have become a more and more important topic. In theory, Chang et al. generalized the Perron-Frobenius Theorem from nonnegative matrices to nonnegative tensors in [1]. Y. Yang and Q. Yang extended their results in [2,3]. The latest result on the Perron-Frobenius Theorem is that the eigenvalues with modulus ρ(A) have the same geometric multiplicity in [4]. Some other results of nonnegative tensors were established in [5][6][7][8][9][10][11][12]. What is more, Ng, Qi, and Zhou proposed the NQZ method for finding spectral radius of a nonnegative irreducible tensor in [13]. Pearson obtained that the NQZ method would converge if the tensor with even order is essentially positive in [14]. In [15] Chang, Pearson and Zhang proved the convergence of the NQZ method for primitive tensors with any nonzero nonnegative initial vector. Zhang and Qi gave the linear convergence of the NQZ method for essentially positive tensors in [16]. Hu, Huang, and Qi [17] established the global R-linear convergence of the modified version of the NQZ method for nonnegative weakly irreducible tensors which were introduced by Friedland, Gaubert, and Han in [18]. Chen, Qi, Yang, et al showed an inexact power-type algorithm for finding spectral radius of nonnegative tensors in [19].
In this paper, we focus on a method with parameter for finding the spectral radius of weakly irreducible nonnegative tensors. What is more, the method has an explicit linear convergence rate for indirectly positive tensors. Interestingly, the algorithm is exactly the NQZ method 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 the inexact power-type algorithm in [19].
This paper is organized as follows. In Sect. 2, we recall some theorems and the NQZ method. And a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors. Then we prove it has an explicit linear convergence rate for indirectly positive tensors. In Sect. 3, the linear convergence rate for the method is established. In Sect. 4, a modified NQZ method is presented and the inexact power-type algorithm is promoted. In Sect. 5, we report some numerical results.
We first add a comment on the notation that is used in this paper. Vectors are written as lowercase letters (x, y, · · · ), italic capitals (A, B, · · · ) are for matrices, and tensors correspond to calligraphic capitals (A, B, · · · ). The entry in a tensor A, (A) i 1 ···i p , j 1 ··· j q = a i 1 ···i p , j 1 ··· j q . R n + (R n ++ ) is for the cone {x ∈ R n |x i (>)0, i = 1, · · · , n}. The symbol A > ( , , <)B denotes that a i j > ( , , <)b i j for every i, j.

Preliminaries
In this section, we first recall some preliminaries knowledge on nonnegative square tensors. Then a method with parameter is proposed for finding the spectral radius of weakly irreducible nonnegative tensors.
Firstly, we recall some known definitions about tensors. Definition 2.1 (Definition of [1]) A tensor is a multidimensional array, and a real m-th order n dimensional tensor A consists of n m real entries: where i j = 1, · · · , n for j = 1, · · · , m. For any vector x and any real number m, denote If there are a complex number λ and a nonzero complex vector x satisfying the following homogeneous polynomial equations: then λ is called an eigenvalue of A and x the eigenvector of A associated with λ, where Ax m−1 is vectors, whose ith component are The spectral radius of tensor A is defined as ρ(A) = max{|λ| : λ is an eigenvalue of A}.

Definition 2.3 ([20]
) An m-th order n dimensional tensor C = (c i 1 · · · c i m ) is called reducible, if there exists a nonempty proper index subset I ⊂ {1, · · · , n} such that If C is not reducible, then we call C irreducible.

Definition 2.4 (Definition 2.2 of [21]) For any vector
We define the composite of the tensors for x ∈ R + to be the function (not necessarily a tensor) (B • A)x = Bω m−1 .  Remark Suppose A is essentially positive tensor. It is easy to obtain G(A) > 0 by Definition 2.8. So essentially positive tensors are indirectly positive tensors, but not vice versa. We will use one example to illustrate it.
Next we recall the NQZ method for an irreducible nonnegative square tensor in [13].
Algorithm 2.12 has the following properties.
To speed up the convergence rate of Algorithm 2.12, we present an algorithm with parameter in a different way as follows. Considering where μ i is parameter. One can get suitable algorithm by taking a specific parameter. Because there is countless viable parameters, it has plenty of room for improvement. We will give one kind of parameters, which get a faster convergence rate in some conditions.

Linear Convergence Analysis for Algorithm 2.15
In this section, we prove Algorithm 2.15 has an explicit linear convergence rate for indirectly positive tensors.
, if x and y are comparable, +∞, otherwise for x, y ∈ R n + \{0}. Definition 3.2 (Definition of [18]) Let f = ( f 1 · · · f n ) T : R n → R n be a polynomial map. We assume that each f i is a polynomial of degree d i 1, and that the coefficient and there exist constant θ ∈ (0, 1) and positive integer M such that holds for all k 1.

Theorem 3.5 Suppose A is an m-th order n dimensional weakly primitive nonnegative tensor. If the sequence {x
Hence the convergence of sequence {x (k) } by Algorithm 2.15 can be reached by the Corollary 3.3. Then this theorem is the same as Theorem 3.4.
n are all homogeneous polynomials of degree 0. So they do not change for the different norms of x. Theorem 3.6 Let A be an m-th order n dimensional weakly irreducible nonnegative tensor. Assume that {λ (k) } and {λ (k) } are two sequences generated by Algorithm 2.15. Then where i k+1 and j k+1 satisfy Similarly, we get And we have This completes the proof.
Remark We give the linear convergence analysis for indirectly positive tensor by this theorem. The condition of indirectly positivity is weaker than the condition of essentially positivity in Lemma 2.14. So this result is more useful in inexact algorithms. Furthermore, this theorem can be extended to find the largest singular values of a rectangular tensor.
Remark We know there exists a > 0 but it is hard to find the explicit number. So in the complexity analysis we always use In fact, we can compute a ε 0 = min i x (k) max i x (k) when ε 0 = 10 −2 . Then we use " a ε 0 " instead of "a" to give a complexity analysis when ε = 10 −7 . In this way, we give the most iterations when ε is from 10 −2 to 10 −7 .
In the following we give a complexity analysis.

Theorem 3.10 If
A is an m-th order n dimensional nonnegative tensor with a ii···i j > 0, i, j = 1, · · · , n, then Algorithm 2.15 terminates in at most Proof By Theorem 3.8, we have for k = 0, 1, · · · , In order to ensureλ we only need Then we have This completes the proof.
Remark Clearly, Algorithm 2.15 coincides with NQZ algorithm when p is one. We know linear convergence rate varies when p changes. One can refer the numerical results in Sect. 5 for the better choice of parameter p.

Linear Convergence Analysis for Algorithm 4.1
In this section, we present a modified NQZ method and we promote the inexact power-type algorithm. We find that the modified NQZ method has an explicit linear convergence rate for nonnegative tensors with any positive initial vector and this method has an error bound for nonnegative tensors with a positive Perron vector.
Because the NQZ method cannot give an explicit convergence rate for finding the spectral radius of a weakly irreducible nonnegative tensor, we present a modified NQZ method for finding the spectral radius of a weakly irreducible nonnegative tensor by a specific perturbation tensor, which can give an explicit convergence rate. Let A 0 be a nonnegative tensor of m-th order n dimensional with a ii···i j = 1, i, j = 1, · · · , n, the other entries being equal to 0. The algorithm is as follows: Step 1 Given i , i = 1, 2, · · · , n and set k = 0.
Step 2 Compute Step . Otherwise, replace k by k + 1 and go to step 2.
Lemma 4.2 (Theorem 2.8 of [24]) Suppose A is an m-th order n dimensional nonnegative tensor with a positive Perron vector and A = A + A is the perturbed nonnegative tensor of A. Then we have

Lemma 4.3 Suppose A is an m-th order n dimensional nonnegative tensor with a positive Perron vector x and
Proof Let x be the Perron vector of A. Then we have Similarly, one has Because A has a positive Perron vector. Then we have ρ(A) ρ( A) and Hence by Lemma 2.7 of [24], we have

Remark Because τ (A) (τ (A)) m−1 , Lemma 4.3 is a more useful conclusion than Lemma 4.2.
Similarly, one has the following:

Lemma 4.4 Suppose
A is an mth order n dimensional nonnegative tensor with a positive Perron vector and A = A + ε A 0 is the perturbed nonnegative tensor of A. Let x be positive Perron vector of A. If x 1 = 1, then Furthermore, if A has a positive Perron vector x, then Proof Because in Algorithm 4.1 B = A+ε 0 A 0 and B is an m-th order n dimensional nonnegative tensor with a ii···i j 1, i, j = 1, · · · , n. Then by Theorem 3.8 both {λ (k) } and {λ (k) } in Algorithm 4.1 converge linearly to ρ(B). In details, for k = 0, 1, · · · , And by Lemma 4.3 we have This completes the proof.
Remark By Theorem 4.5 we can find that Algorithm 4.1 has an explicit linear convergence rate for weakly irreducible nonnegative tensors and Algorithm 4.1 has an error bound for nonnegative tensors with a positive Perron vector.

Theorem 4.6 If
A is an m-th order n dimensional nonnegative tensor with a ii···i j > 0, i, j = 1, · · · , n, then Algorithm 4.1 terminates in at most Proof The proof is similar to that of Theorem 3.10, so we omit it.
In the following, we promote the inexact power-type algorithm. We first recall the inexact power-type algorithm for finding spectral radius of nonnegative tensor A in [19]. Let E be the all-ones m-th order n dimensional tensor.

Numerical Results
In this section, we give the numerical results and give a comparison for convergence rate between Algorithm 4.1 and the algorithm in [24].
When p = 1, Algorithm 2.15 is exact the NQZ method. Actually, for the different choices of p, Algorithm 2.15 has different performances. We state two examples to illustrate this result, and compare the performances for p = 0.5, 0.9, 1, 1.1, 2.
Example 5.1 We consider a 3-th order 3 dimensional nonnegative tensor A and B, where a 132 = a 231 = 2, a 312 = 1 and b 122 = b 213 = 2, b 211 = 1, b 333 = 3. It can be shown that the tensor A is weakly primitive, while the tensor B is weakly reducible. We can compute the spectral radius by Algorithm 2.15 for different p within 1 000 iterations. we show the convergence result for each p, respectively. It is easy to see the graph of G(A) that the tensor A is not indirectly positive but A is a primitive tensor, and thus we can learn that NQZ method is not global linear convergent. The experiment shows our correctness of this conclusion.
Example 5.3 We state the same example at (Example 3 [24]) to illustrate our tighter bound to compute the spectral radius of the tensor A, where a 122 = a 133 = a 211 = a 311 = 1 and the other entries are equal to zero. The spectral radius of A is equal to √ 2 ≈ 1.414 213 562 373 095, see [15]. It is obvious that tensor A is irreducible but not primitive. Thus the NQZ algorithm for such A will be not convergent. We use Algorithm 4.1 to test this example and use the same parameters.
We first show the result gained from (Example 22 [24]), A = A + εE, where E is a tensor with all the entries being equal to one. Then we state our result as follows: A = A + ε A 0 , where A 0 is a nonnegative tensor of order m and dimension n with a ii···i j = 1, i, j = 1, · · · , n, the other entries are equal to 0. It is clear that Algorithm 4.1 is more sufficient than that in [24].
What is more, we compare these two Algorithms as follows under the same error bound. It is clear that our Algorithm needs less iterations.
Example 5. 4 We present another example to illustrate our tighter bound to compute the spectral radius of the tensor A. Let A be a 10-th order 4 dimensional nonnegative tensor, where a i j··· j = 1 i , 1 i, j 4, and the other entries are equal to zero. The spectral radius of A is equal to 25 12 ≈ 2.083 333 333 33. We use Algorithm 4.1 to test this example and use the parameters of 1 .
We first show the result gained with the perturbation tensor in ( [24]), A = A + εE, where E is a tensor with all the entries being equal to one, ε = 10 −6 : Then we state our result as follows: A = A + ε A 0 , where A 0 is a nonnegative tensor of order m and dimension n with a ii···i j = 1, i, j = 1, · · · , n, the other entries are equal to 0, ε = 10 −6 : It is clear that our algorithm is more efficient than that in [24].
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