An algorithm for calculating spectral radius of s -index weakly positive tensors

: In this paper, we introduced s -index weakly positive tensors and discussed the calculation of the spectral radius of this kind of nonnegative tensors. Using the diagonal similarity transformation of tensor and Perron-Frobenius theory of nonnegative tensor, the calculation method of the maximum H -eigenvalue of s -index weakly positive tensors was given. A variable parameter was introduced in each iteration of the algorithm, which is equivalent to a translation transformation of the tensor in each iteration to improve the calculation speed. At the same time, it was proved that the algorithm is linearly convergent for the calculation of the spectral radius of s -index weakly positive tensors. The final numerical example shows the e ff ectiveness of the algorithm.


Introduction
Consider an m-order n-dimensional square tensor A consisting of n m entries in the real field R: then A is called an m-order n-dimensional nonnegative tensor.Denote the set of all m-order n-dimensional nonnegative tensors as R [m,n]   + .R n , R n + , R n ++ represents the set of all n-dimensional vectors, the set of all n-dimensional nonnegative vectors and the set of all n-dimensional positive vectors, respectively.Tensors play an important role in physics, engineering and mathematics.There are many application domains of tensors such as data analysis and mining, information science, image processing and computational biology [1][2][3][4].
In 2005, Qi [2] introduced the notion of eigenvalues of higher-order tensors and studied the existence of both complex and real eigenvalues and eigenvectors.Independently, in the same year, Lim [3] also defined eigenvalues and eigenvectors but restricted them to be real.
Qi [2] proposed the definition of H-eigenvalue.If a real number λ and a nonzero real vector x ∈ R n satisfy the following homogeneous polynomial equation: where 1≤i≤n is an n-dimensional vector and (x [m−1] ) i = x m−1 i , then λ is an H-eigenalue of A and x is an H-eigenector of A associated with λ.Define the set of H-eigenvalues of A ∈ R [m,n]   + as σ H (A) and ρ(A) = max λ∈σ H (A) |λ| as the H-spectral radius of tensor A.
Let σ(A) be the set of eigenvalues of a tensor A.

+
. Ng et al. [5] proposed the NQZ algorithm for the largest H-eigenalue of a nonnegative irreducible tensor.
Subsequently, the NQZ algorithm was proved to be convergent for primitive tensors in [6] and for weakly primitive tensors in [4], and the NQZ algorithm was shown to have an explicit linear convergence rate for essentially positive tensors in [7].However, some examples [5] showed that it did not converge for some irreducible nonnegative tensors.
In 2010, Liu et al. [8] modified the NQZ algorithm and proposed the LZI algorithm.Let E = (δ i 1 i 2 •••i m ) be the m-order n-dimensional unit tensor whose entries are
Step 1. Compute , replace k by k + 1 and go to Step 1.
Liu et al. [8] proved that the LZI algorithm is convergent for irreducible nonnegative tensors.In 2012, Zhang et al. [9] proved the linear convergence of the LZI algorithm for weakly positive tensors.Since then, there have been many studies on the calculation of the maximum eigenvalue of nonnegative tensors.For example, Yang and Ni [10] gave a nonlinear algorithm for calculating the maximum eigenvalue of symmetric tensors; another example, Zhang and Bu [11] gave a diagonal similar iterative algorithm for calculating the maximum H-eigenvalue of a class of generalized weakly positive tensors.

Preliminaries
In this section, we mainly introduce some related concepts and important properties of tensors and matrices.For a positive integer n, let ⟨n⟩ = {1, 2, . . ., n}.Definition 2.1.[12] An m-order n-dimensional tensor A is called reducible if there exists a nonempty proper index subset I ⊂ ⟨n⟩ such that Definition 2.4.[9] Let A be a nonnegative tensor of order m and dimension n.A is weakly positive if a i j••• j > 0 f or i j and i, j ∈ {1, 2, • • • , n}.
(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.
+ and π s−1 (i, j) be an arrangement of s − 1 letters i and m − s letters j.If there exists s ∈ ⟨m − 1⟩ and i 0 ∈ ⟨n⟩ for any j ∈ ⟨n⟩, j i 0 , such that a i 0 π s−1 (i 0 , j) 0 and a jπ s−1 ( j,i 0 ) 0 hold, then A is called an s-index weakly positive tensor.
For example, A = (a i jk ) ∈ R [3,3]  + , where elsewhere, then A is a two-index weakly positive tensor.
Remark 2.1.The essentially positive tensors, the weakly positive tensors and the generalized weakly positive tensors are all one-index weakly positive tensors, which are special tensor classes of s-index weakly positive tensors..

Theorem 2.1. [15] For any nonnegative tensor
Theorem 2.3.[12] If A is an irreducible nonnegative tensor of order m and dimension n, then there exists λ 0 > 0 and x 0 > 0,

+
. The tensors A and B are said to be diagonal similar if there exists some invertible diagonal matrix D Theorem 2.4.[17] If the two m-order n-dimensional tensors A and B are diagonal similar, then σ(A) = σ(B).

An algorithm and its convergence analysis
In 1981, Bunse [18] gave a diagonal similar iterative algorithm for calculating the maximum eigenvalue of irreducible nonnegative matrices.In 2008, Lv [19] further studied the diagonal similarity iterative algorithm for calculating the maximum eigenvalue of irreducible nonnegative matrices.In 2021, Zhang and Bu [11] gave a diagonal similar iterative algorithm for calculating the maximum H-eigenvalue of nonnegative tensors.In this paper, according to the construction idea of the algorithm in [19], a numerical algorithm for calculating the maximum H-eigenvalue and corresponding eigenvector of s-index weakly positive tensors is given.
Step 3. Set and replace k by k + 1, go to Step 1.
In the following, we will give the convergence condition of Algorithm 3. Define
+ be an s-index weakly positive tensor, then it is an irreducible tensor.
Proof.If a jπ s−1 ( j,i 0 ) 0, then a (k) In the case of j i 0 , it can be obtained from Lemma 3.1 that where ã = min Then, by r(A (0) ) ( j i 0 ) can be obtained by (3.1).In the case of j = i 0 , it holds that a (k) + be an s-index weakly positive tensor.That is, there are s ∈ ⟨m − 1⟩ and i 0 ∈ ⟨n⟩ for any j ∈ ⟨n⟩, j i 0 , such that a i 0 π s−1 (i 0 , j) 0 and a jπ s−1 ( j,i 0 ) 0 hold, then for Algorithm 3, Proof.Let A [0] = (a [0]  i j ) n×n , a i 0 j = a i 0 π s−1 (i 0 , j) , a ji 0 = a jπ s−1 ( j,i 0 ) , j ∈ ⟨n⟩ and zero elsewhere, then , ε} =: â.Thus, where α = 1 − â 2r(A (0) ) .Therefore, Note that 0 < α = 1 − â 2r(A (0) ) < 1, and we can obtain lim If A is an s-index weakly positive tensor, then by Algorithm 3 there must be From Definitions 2.3-2.5, we can see that the essentially positive tensors, the weakly positive tensors and the generalized weakly positive tensors are all s-index weakly positive tensors, so we have the following corollary.that is, It can be seen from Theorem 2.4 that r i ( Â) = ρ(A), i ∈ ⟨n⟩; therefore ρ(A)( De) m−1 = A( De) [m−1] ,

Numerical examples
In this section, to show the effectiveness of Algorithm 3, we compare it with the LZI algorithm.For the parameter α k in the algorithm, we selected different values and compared the corresponding results.
(2) The number of iteration exceeds 10 4 .Some numerical results are given in Table 1, where ρ(A) denotes the H-spectral radius of A, Iter denotes the iteration of the algorithms and Time(s) denotes the CPU time (in seconds) used when the conditions (1) are met.
Table 1 shows a comparison between Algorithm 3 and the LZI algorithm given in [8], with the same error and the number of iterations and calculation time significantly reduced, which further verifies that our proposed algorithm is more efficient.+ (m = 3), whose all entries of random values drawn from the standard uniform distribution on (0, 1).
Obviously, this is an s-index weakly positive tensor (s = 1 or 2).Choose ε = 10 −8 and the termination conditions are the same as in Example 4.1.Take different values for α k in Algorithm 3, and the corresponding results are shown in Table 2, where r − r = r(A (k) ) − r(A (k) ).From the data in Table 2, it can be seen that when the value of α k is different, there are differences in the number of iteration steps and operation times.The calculation time is almost the same, but the difference in iteration steps is quite significant, so selecting the appropriate α k will improve the efficiency of the algorithm.

Conclusions
In this paper, a class of s-index weakly positive tensors was defined and a diagonal similar iterative algorithm for the maximum H-eigenvalue of such tensors was given.In the algorithm, a variable parameter was introduced in each iteration, which is equivalent to a translation transformation for each iteration of the tensor.Compared with the LZI algorithm, the number of iterations and time of calculation have great advantages.It was also proved that the algorithm has linearly convergence for s-index weakly positive tensors.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Table 1 .
The comparison of the Algorithm 3 and LZI algorithm.

Table 2 .
The comparison of different values of α k in Algorithm 3.