1 Introduction

In recent years, tensor eigenvalue problems have gained increasing attention due to their broad applicability across various scientific and engineering domains. Notable examples include the best rank-one tensor approximation in data analytics [1,2], modeling of higher-order Markov chains [3], studies in solid mechanics and quantum entanglement [4,5], and structural analysis of multilayer networks [6,7]. The concept of tensor eigenvalues was independently introduced by Qi [8] and Lim [9] in 2005, marking a significant advance in tensor analysis. Building upon this foundation, Ng, Qi and Zhou [3] proposed the NQZ algorithm in 2009 to compute the H-spectral radius of irreducible nonnegative tensors. This algorithm serves as a fundamental tool in the fields of tensor spectral theory and numerical multilinear algebra, and is widely regarded as a natural extension of the classical power method for calculating the dominant eigenvalue of matrices. A substantial body of work has since been devoted to the analysis and development of the NQZ algorithm and its variants, leading to important theoretical and algorithmic advancements [3,10–17]. For example, Pearson [11] proved the convergence of the NQZ algorithm for essentially positive tensors. Furthermore, Liu, Zhou, and Ibrahim [13] examined a variant-referred to as the LZI algorithm and established its convergence for a class of primitive tensors.

Further progress has been made in understanding the convergence behavior of the NQZ algorithm under broader and more general conditions. In 2011, Chang, Pearson, and Zhang [18] established the convergence of the NQZ algorithm for primitive tensors. Building on this work, Zhang and Qi [12] later proved the linear convergence of the NQZ algorithm for essentially positive tensors, and subsequently demonstrated the linear convergence of an improved variant known as the LZI algorithm for weakly positive tensors [14]. In particular, both analyses were performed under highly restrictive conditions, relying on the structural properties of the majorization matrix associated with a nonnegative tensor, which was required to be either a fully positive matrix or to have all positive off-diagonal entries. In 2014, Hu, Huang, and Qi [17] further advanced this line of research by proving the R-linear convergence of the NQZ algorithm for weakly primitive tensors. However, their analysis did not provide an explicit range or upper bound for the convergence factor R, limiting its practical applicability. Despite these constraints, these foundational works significantly enriched the theoretical development of the NQZ algorithm within the field of nonnegative tensor spectral analysis.

Subsequent studies have continued to broaden the scope of linear convergence analysis for algorithms computing the H-spectral radius of nonnegative tensors. For instance, in 2021, Zhang and Bu [16] introduced a diagonal similarity-based algorithm tailored for a newly defined class of weakly positive tensors and established its linear convergence. More recently, in 2024, Liu and Lv [10] extended the concepts of essentially positive, weakly positive, and generally weakly positive tensors by introducing the notion of weakly essentially irreducible nonnegative tensors. They proposed a corresponding algorithm for computing the H-spectral radius and established more general conditions ensuring its linear convergence. In parallel with these advances, alternative computational frameworks have also been developed. For example, Gautier, Tudisco, and Hein [19] introduced the concept of -eigenvalue for nonnegative tensors and designed a general algorithm applicable to weakly irreducible cases. In particular, when the resulting -spectral radius coincides with the classical H-spectral radius, thus unifying and extending existing computational frameworks.

Despite these advances, the development of a unified and less restrictive convergence theory remains an open challenge. In this paper, we contribute to closing this gap by establishing the R-linear convergence of the NQZ algorithm from a novel perspective via the directed graph associated with a nonnegative tensor. Leveraging the structural characteristics of tensor-induced directed graphs, we not only derive an explicit upper bound for the convergence factor R, but also propose a more general sufficient condition for its linear convergence. This condition substantially relaxes the stringent assumptions imposed in previous works such as [12,14], thus broadening the theoretical foundation and enhancing the practical applicability of the NQZ algorithm.

This paper is organized as follows. Sect 2 provides a review of the relevant background and introduces the NQZ algorithm. In Sect 3, we establish the R-linear convergence of the NQZ algorithm for computing the H-spectral radius of a class of weakly irreducible nonnegative tensors by utilizing their associated directed graphs.

We also present more general conditions ensuring the linear convergence of the algorithm. Sect 4 summarizes the main findings of this work and discusses potential directions for future research.

2 Preliminaries

An mth-order tensor with dimension n over the real numbers is a multi-way array consisting of n^m real-valued entries, represented as

where . In the special case when m = 2, the tensor reduces to an matrix. A tensor is termed nonnegative if every entry satisfies . The set of all real-valued tensors of order m and dimension n is denoted by , while designates the subset comprising nonnegative tensors. Correspondingly, denotes the space of all real n-dimensional vectors, with and representing the sets of nonnegative (including zero) and strictly positive vectors, respectively. Furthermore, denotes the set of all real matrices.

In 2005, the concept of eigenvalues for tensors was independently introduced by Lim [8] and Qi [9].

Definition 2.1. Consider a real tensor of order m and dimension n, i.e., . A complex scalar is called an eigenvalue of if there exists a non-zero vector , such that

where the vector is defined component-wise as

and the vector is given by

In this setting, x is referred to as the eigenvector corresponding to the eigenvalue λ.

The spectral radius of the tensor , denoted by , is defined as the supremum of the absolute values of all its eigenvalues, i.e.,

where denotes the spectrum of .

In particular, if both the eigenvalue λ of and its corresponding eigenvector x are real, i.e., and , then the pair is called an H-eigenpair, and λ is referred to as an H-eigenvalue of .

In 2008, Chang et al. [20] extended the classical notion of irreducibility from matrices to tensors.

Definition 2.2. ([20]) An m-th-order, n-dimensional tensor is said to be reducible if there exists a nonempty proper subset such that

If no such subset J exists, then is called irreducible.

Definition 2.3. Let . We recall several important notions related to :

(1) ([11]) The majorization matrix associated with is the nonnegative matrix whose (i,j)-th entry is given by

(2) ([11,14]) The tensor is called essentially positive if for every . It is said to be weakly positive if for all pairs with .

(3) ([16]) The tensor is termed generalized weakly positive if there exists an index such that for all ,

The works presented in Refs [12,14,16] focus on the development of algorithms and the analysis of linear convergence for the H-spectral radius associated with essentially positive tensors, generalized weakly positive tensors, and weakly positive tensors, respectively. Moreover, in 2014, Hu et al. [17] introduced an equivalent characterization of weakly irreducible tensors, further enriching the theoretical framework in this area.

Definition 2.4. Let be a nonnegative tensor.

(1) ([17]) The representation matrix associated with is defined as the nonnegative matrix whose (i,j)-th entry is given by the sum of all entries for which at least one of the indices equals j.

(2) ([17]) The tensor is said to be weakly reducible if the matrix is reducible; otherwise, is weakly irreducible. Furthermore, is weakly primitive if is a primitive matrix.

(3) ([15]) The tensor is called indirectly positive if is strictly positive, and indirectly weakly positive if is strictly positive, where I denotes the identity matrix.

Furthermore, Chang et al. [20] extended the classical Perron-Frobenius theorem from nonnegative matrices to nonnegative tensors.

Theorem 2.1 ([20]) Let be a nonnegative tensor of order m and dimension n. Then the following statements hold:

(i) There exist a scalar and a nonnegative vector such that

(ii) If is weakly irreducible, then and the associated eigenvector is strictly positive, that is, . Moreover, is the unique eigenvalue corresponding to a nonnegative eigenvector, and every eigenvalue λ of satisfies .

Based on statement (ii) of Theorem 2.1, it follows that the H-spectral radius of a nonnegative tensor is itself an eigenvalue and .

In 2010, Yang et al. [21] further generalized the classical bounds on the spectral radius from nonnegative matrices to nonnegative tensors.

Theorem 2.2. ([21]) Let be a nonnegative tensor and denote by its H-spectral radius. Then the following inequalities hold:

Consider a nonnegative tensor . It can be represented by a directed graph , where the set of vertex is . A directed edge (i,j) belongs to if there exist indices such that and the tensor entry is non-zero. A walk from vertex i to vertex j in is a sequence of vertices with each consecutive pair being an edge in for . If i = j, such a walk is called a non-simple path. When all vertices are distinct, γ is referred to as a directed path connecting i and j. The graph is said to be strongly connected if for every pair of distinct vertices i and j, there exists a directed path from i to j.

Based on this, Friedland et al. [22] introduced the concept of weak irreducibility for nonnegative tensors as follows:

Definition 2.5. ([22]) An m-th order n-dimensional nonnegative tensor is called weakly irreducible if its associated directed graph is strongly connected.

In 2009, Ng et al. [3] proposed the NQZ method for the largest H-eigenvalue of a nonnegative irreducible tensor.

Algorithm 1. ([3]) NQZ algorithm.

Step 0. Choose Let and set .

Step 1. Compute

Step 2. If , stop. Otherwise, replace k by k + 1 and go to Step 1.

In this paper, for , we define , represents a permutation of the nodes .

According to the NQZ algorithm, denoted as , , we can obtain

3 Linear convergence of the NQZ algorithm

In this section, we establish the R-linear convergence of the NQZ algorithm by using the structural properties of directed graphs of tensors. Additionally, we derive an upper bound for the convergence factor R, which is associated with the directed paths of directed graphs of nonnegative tensors. Furthermore, we provide more generalized conditions for the linear convergence of the NQZ algorithm based on the parameters of the upper bound expression for R.

Let . We define the set of all simple paths from node i to node j in the graph as

where denotes the length of the path

Additionally, define

and

Next, we demonstrate that during the computation of the H-spectral radius of a weakly irreducible nonnegative tensor using the NQZ algorithm, a consistent, nonzero, and nonnegative lower bound exists for all nonzero elements throughout the algorithm’s iterations.

Lemma 3.1. ([3]) Let , be the H-spectral radius of . Then, we have monotonically increasing converges to , monotonically decreasing converges to , and

The proof process can be found in Theorem 2.4 of [3].

Lemma 3.2. Let be weakly irreducible. Then for the NQZ algorithm, we have

(1)

where , .

Proof. By displacement algorithm, we know Without loss of generality, assume

(2)

Since the tensor is weakly irreducible, according to Definition 2.2, there exists a directed path γ from vertex t₁ to vertex t_n in the associated digraph , where , − such that > 0. Therefore, combining Lemma 3.1, (2) and we have

i.e.,

(3)

Similarly, we get

(4)(5)(6)

So by (3) and (4), we obtain

that is,

(7)

By (7) and (5), we can get

that is,

Following this sequence of steps, we obtain

(8)

When , it can be obtained from equation (8) that

that is,

When m = 2, it can be obtained from equation (8) that

This completes the proof.

In 2014, Hu et al. [17] proved the R-linear convergence of H-spectral radius NQZ algorithms for weakly primitive tensors. In contrast to their work, we establish the R-linear convergence of the NQZ algorithm for a class of weakly irreducible nonnegative tensors from a different perspective, employing directed graphs of tensors. Additionally, we derive an upper bound on the root convergence factor R and provide a general condition for the linear convergence of the NQZ algorithm.

Theorem 3.1. Let be weakly irreducible. Define

if there exists such that then for the NQZ algorithm, when , the following inequality holds:

Furthermore, for ,

it follows that the NQZ algorithm exhibits R-linear convergence, where the convergence factor α satisfies , and

(9)

The value of is shown in the proof.

Proof. Only the proof for is given, and the proof for m = 2 is similar.

Assuming , otherwise if , we can obtain from Theorem 2.2, and then .

From the NQZ algorithm, it follows that

(10)

(I) Take , and assume that .

(i) Since , there exists a permutation such that . For i₀, by Equation (10), we obtain

where satisfying , .

Applying the above equation, a similar result can be obtained

And then, similarly to the discussion above, we have

Similar, when , then we have

When , then we obtain

a similar discussion leads to

(ii) For any , since is weakly irreducible, we know that is strongly connected, and so there exist , where are not the same as each other, and . There are two cases:

(1) When , we have

By applying the previous equation in sequence, we ultimately obtain

(2) When , a discussion similar to (1) leads to

Combining (i) and (ii) by Lemma 3.1, for any , there are always

or

Take , then there are

Denote , then , there are

(11)

For any , there exists a positive integer such that kr₀. This implies that , by (7), we have

By

we know that 0 < R < 1. Therefore, the NQZ algorithm is R-linearly convergent, and the root convergence factor .

Applying Lemma 3.1 there are

applying Theorem 2.1 again we have .

(II) When , or , assume that , then we have

where .

Thus, a discussion similar to that of (I) leads to the conclusion.

Remark 3.1. Using Theorem 3.1, we can prove that the NQZ algorithm is R-linearly convergent and provide an upper bound on its root convergence factor, which. is , where the value of α is shown in equation (9).

Remark 3.2. For a general weakly irreducible nonnegative tensor , it suffices to define , where . This ensures that satisfies the conditions of Theorem 3.1.

Applying Theorem 3.1, we obtain

Theorem 3.2. Let be weakly irreducible. If for any , it holds that , then for a given , when , applying the NQZ algorithm, there must be .

More general conditions for the linear convergence of the NQZ algorithm can be easily derived from Theorem 3.1.

Theorem 3.3. Let be weakly irreducible. If there exists an index , such that , and for any , then by applying the NQZ algorithm,

where the value of is given in Theorem 3.1.

Proof. Since for any , we have , therefore,

Thus, the conclusion follows from Theorem 3.1 and its proof.

Remark 3.3. Theorem 3.2 establishes more general conditions for the linear convergence of the NQZ algorithm in computing the H-spectral radius of a nonnegative tensor, based on the directed graph associated with the tensor.

Literature in Refs [12,14] demonstrates that the NQZ algorithm achieves linear convergence when computing the H-spectral radius of essentially positive tensors or weakly positive tensors. The following example demonstrates that the condition proposed in this article for the linear convergence of the NQZ algorithm, as stated in Theorem 3.2, provides a more comprehensive and generalized framework compared to the results presented in [12,14].

Example 3.1. Let , where , the remaining elements are zero.

The majorization and representation matrices of are respectively

Clearly, by Definition 2.3, we know that is not an essentially positive tensor or a weakly positive tensor. The linear convergence result of the NQZ algorithm cannot be derived from the literature [12,14]. Additionally, the tensor is neither a generalized weakly positive tensor nor an indirectly positive tensor, which means it does not meet the conditions for linear convergence outlined in [15,16]. But by , we know that r₀ = 1, and that there is . Thus, the NQZ algorithm is linearly convergent by Theorem 3.2.

Example 3.2. Let , where , , for a certain ,, the remaining elements are zero.

If we take j₀ = 1, then the optimization matrix , and the representation matrix has the following form

where * represents non-zero elements. Clearly, by Definition 2.3, we know that is not an essentially positive tensor or a weakly positive tensor. The linear convergence result of the NQZ algorithm cannot be derived from the literature [12,14]. But by , we know that r₀ = 1, and that there is . Thus, the NQZ algorithm is linearly convergent by Theorem 3.2.

In Example 3.2, with , and j₀ = 1, the NQZ algorithm is applied to compute the H-spectral radius in Examples 3.1 and 3.2, and its linear convergence is shown in Fig 1.

4 Conclusion

In this paper, we explore the linear convergence of the NQZ algorithm for calculating the H-spectral radius of a nonnegative tensor. By utilizing the directed graph of a tensor, we demonstrate that the NQZ algorithm exhibits R-linear convergence for a specific class of weakly irreducible nonnegative tensors (Theorem 3.1). We establish an upper bound for the root convergence factor R and provide a general condition for the linear convergence of the NQZ algorithm (Theorem 3.2).

Acknowledgments

The authors are very grateful to the reviewers for their valuable comments that improved the manuscript.