Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors

Hongbin Lv; Meixiang Chen

doi:10.1515/math-2025-0194

Article Open Access

Algorithm and linear convergence of the H-spectral radius of weakly irreducible quasi-positive tensors

Hongbin Lv and Meixiang Chen

Published/Copyright: September 19, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

A class of weakly irreducible quasi-positive tensors is defined by using directed hypergraphs of tensors, which generalizes the essential positive tensors, weakly positive tensors, generalized weakly positive tensors, and weakly essential irreducible nonnegative tensors. Furthermore, an algorithm based on displacement transformation for computing the H -spectral radius of such weakly irreducible quasi-positive tensors is discussed. The R -linear convergence of the proposed algorithm is established, and the conditions are also provided to ensure its linear convergence.

Keywords: quasi-positive tensor; weakly irreducible; H-spectral radius; algorithm; linear convergence

MSC 2010: 15A69; 65F10

1 Introduction

An m th-order n -dimensional real tensor is a multidimensional array comprising n m elements of the form

A = ( a j 1 j 2 … j m ) , a j 1 j 2 … j m ∈ R , j i ∈ ⟨ n ⟩ , i ∈ ⟨ m ⟩ ,

where ⟨ n ⟩ = { 1 , 2 , … , n } , ⟨ m ⟩ = { 1 , 2 , … , m } . If m = 2 , A reduces to an n × n matrix. A tensor A is nonnegative if each entry is nonnegative. We denote the set of all real m th-order n -dimensional tensors by R [ m , n ] , and the set of all m th-order n -dimensional nonnegative tensors by R + [ m , n ] . And denote the set of n -dimensional real vectors, nonnegative nonzero vectors, and positive vectors as R n , R + n , R + + n respectively. Let C n be the set of n -dimensional complex vectors, and D + n be the set of n × n positive diagonal matrices.

Nonnegative tensors, as significant extensions of nonnegative matrices, have found successful applications in many fields such as signal processing, magnetic resonance imaging, data mining, and neuroscience [1–3].

The m th-order n -dimensional identity tensor, denoted by ℐ = ( δ j 1 j 2 … j m ) ∈ R [ m , n ] , is the tensor with entries,

δ j 1 j 2 … j m = 1 , if j 1 = j 2 = … = j m , 0 , otherwise .

For an n -dimensional vector x = ( x 1 , x 2 , … , x n ) T , we define n -dimensional vector A x m − 1 and x [ m − 1 ] , whose j th component is

( A x m − 1 ) j = ∑ j 2 , … , j m = 1 n a j j 2 … j m x j 2 … x j m ,

and

( x [ m − 1 ] ) j = x j m − 1 ,

respectively.

Let A ∈ R [ m , n ] and λ ∈ C . If there is a nonzero vector x ∈ C n such that

A x m − 1 = λ x [ m − 1 ] ,

then λ is called an eigenvalue of A and x an eigenvector of A corresponding to the eigenvalue λ . If the eigenvector x is real, then the eigenvalue is necessarily real as well, and λ and x are called an H -eigenvalue and an H -eigenvector of A , respectively. The definition was initially introduced and extensively studied by Qi [4] and Lim [5], respectively. The largest modulus of the eigenvalues of A , denoted by ρ ( A ) , is called the spectral radius of A . Furthermore, the set of all the eigenvalues of A is referred to as the spectrum of A , denoted as σ ( A ) .

In 2005, Qi [4] and Lim [5] independently defined the eigenvalues of the tensors. In 2009, an algorithm for the H -spectral radius of irreducible nonnegative tensors (known as the NQZ Algorithm) was proposed in [6]. The convergence of this algorithm for essentially positive tensors was proved in [7]. In 2010, the LZI Algorithm was introduced in [8] and its convergence for irreducible nonnegative tensors was established. Further progress was made in 2012, when the linear convergence of the NQZ Algorithm on essential positive tensors was proved in [9], and the linear convergence of the LZI Algorithm on weakly positive tensors was demonstrated in [10]. In 2021, a definition of a generalized weakly positive tensor and an algorithm for the H -spectral radius were presented in [11], along with a proof of the algorithm’s linear convergence. The convergence of the NQZ Algorithm with a primitive tensor was proved in [12]; moreover, its convergence to the weakly primitive tensor was discussed in [13]. Subsequently, the NQZ Algorithm and nonlinear algorithm for the H -spectral radius of general nonnegative tensors have attracted attention. However, the nonlinear algorithm initially requires quantifying the nonnegative tensor as a symmetric tensor with equal H -spectral radius. Recently, a power function-based algorithm for calculating the H -spectral radius of weakly irreducible nonnegative tensors has been proposed in [14]. This algorithm addressed the challenge of calculating the H -spectral radius of general weakly irreducible nonnegative tensors.

In this article, a class of weakly irreducible quasi-positive tensors is defined using directed hypergraphs of tensors, and the algorithm with displacement transformation for the H -spectral radius of weakly irreducible quasi-positive tensors is discussed. This work is organized as follows. In Section 1, we introduce the definition of eigenvalues of a tensor and the related algorithm to compute the H -spectral radius of a nonnegative tensor. In Section 2, we present the properties associated with the H -spectral radius of the nonnegative tensor and review the classical NQZ Algorithm. In Section 3, we introduce the definition of a weakly irreducible quasi-positive tensor by defining a simple directed cyclic path generated by the isometric elements of the directed graph of the tensor, and provide an algorithm (Algorithm 4) for computing its H -spectral radius. In Section 4, we prove the convergence of Algorithm 4 to compute the H -spectral radius of a weakly irreducible quasi-positive tensor and further consider the conditions for the linear convergence of Algorithm 4. Finally, we illustrate the superiority of Algorithm 4 by comparing it with a few concrete examples in Section 5.

2 Preliminaries

In this section, we clarify some notations and properties related to tensors and matrices.

In 2008, Chang et al. [15] extended the concept of irreducible matrices to irreducible tensors.

Definition 2.1

[15] An m th order n -dimensional tensor A is called reducible if there exists a nonempty proper index subset J ⊂ ⟨ n ⟩ such that

a j 1 j 2 … j m = 0 , ∀ j 1 ∈ J , ∀ j 2 , … , j m ∉ J .

If A is not reducible, then A is irreducible.

Moreover, Chang et al. [15] generalized the Perron-Frobenius Theorem for nonnegative matrices to nonnegative tensors.

Theorem 2.1

[15] If A is an mth order n-dimensional nonnegative tensor, then there exist λ 0 ≥ 0 and a nonnegative vector x 0 ∈ R + n such that

(2.1) A x 0 m − 1 = λ 0 x 0 [ m − 1 ] .

Theorem 2.2

[15] If A is an mth order n-dimensional nonnegative irreducible tensor, then

λ 0 > 0 is an eigenvalue;
x 0 > 0 , i.e., all entries of x 0 are positive;
if λ is an eigenvalue with a nonnegative eigenvector, then λ = λ 0 . In addition, the nonnegative eigenvector is unique up to the multiplicative constant;
if λ is an eigenvalue of A , then ∣ λ ∣ ≤ λ 0 .

From the results of assertions (2) and (4) in Theorem 2.2, it is evident that the spectral radius of a nonnegative tensor also constitutes an eigenvalue and is commonly referred to as the H -spectral radius of the nonnegative tensor A , denoted as ρ ( A ) .

In 2010, Yang and Yang [16] extended the classical result of upper and lower bounds of the spectral radius of nonnegative matrices to encompass nonnegative tensors.

Theorem 2.3

[16] Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] . ρ ( A ) is the H-spectral radius of A . Then,

(2.2) min j ∈ ⟨ n ⟩ ∑ j 2 , … , j m = 1 n a j j 2 … j m ≤ ρ ( A ) ≤ max j ∈ ⟨ n ⟩ ∑ j 2 , … , j m = 1 n a j j 2 … j m .

In 2013, Friedland et al. [13] introduced the concept of weakly irreducible nonnegative tensors, leveraging the strong connectivity of graphs.

Given a nonnegative tensor A = ( a j 1 … j m ) ∈ R + [ m , n ] , it is linked to a directed graph G ( A ) = ( V , E ( A ) ) , where V = ⟨ n ⟩ and a directed edge ( j , l ) ∈ E ( A ) exists if and only if there are indices { j 2 , … , j m } such that l ∈ { j 2 , … , j m } and a j j 2 … j m ≠ 0 . A walk in G ( A ) = ( V , E ( A ) ) from vertex j to vertex l is a sequence γ of vertices j = j 0 , j 1 , … , j r = l such that ( j t , j t + 1 ) ∈ E ( A ) is a directed edge for each t = 0 , 1 , … , r − 1 . If the vertices j 0 , j 1 , … , j r are distinct, then γ is a directed path that joins j and l . A directed graph is strongly connected provided that for each pair j and l of distinct vertices, there exists a directed path from j to l .

Definition 2.2

[15] An m th order n -dimensional tensor A is called weakly irreducible if the associate-directed graph G ( A ) is strongly connected.

In 2014, Hu et al. [17] provided an equivalent definition of the weakly irreducible tensor.

Definition 2.3

[17] Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] .

We call a nonnegative matrix G ( A ) as representation associated with nonnegative tensor A , if the ( j , l ) th element of G ( A ) is defined as the summation of a j j 2 … j m over indices l ∈ { j 2 … j m } .
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.

From Definitions 2.2 and 2.3, it is evident that weakly irreducible nonnegative tensors constitute a subclass of nonnegative tensors that are weaker than irreducible nonnegative tensors.

The algorithms and linear convergence properties of the H -spectral radius for the weakly positive tensor, generalized weakly positive tensor, and the essentially positive tensor, respectively, are discussed in [9–11].

Definition 2.4

[7] A nonnegative matrix A ˆ is called the majorization associated with the nonnegative tensor A , if the ( j , l ) th element of A ˆ is defined as a j l … l for any j , l ∈ ⟨ n ⟩ .

Definition 2.5

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] . A ˆ = ( a j l ) ∈ R + n × n is the majorization matrix of A .

[7, 9] A tensor A is essentially positive if a j l > 0 , j , l ∈ ⟨ n ⟩ . A tensor A is weakly positive if a j l > 0 , j ≠ l , j , l ∈ ⟨ n ⟩ .
[11] A tensor A is generalized weakly positive if there exists j 0 ∈ ⟨ n ⟩ , such that a j 0 l … l > 0 , a l j 0 … j 0 > 0 , for all l ∈ ⟨ n ⟩ \ { j 0 } , l ≠ j 0 .

In 2023, the weakly essentially irreducible nonnegative tensor has been defined in [18], and an algorithm is given for its H -spectral radius and conditions for linear convergence.

Definition 2.6

[18] Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] , we call F s ( A ) a matrix of s -index of tensor A , if F s ( A ) = ( a j l ) ∈ R + n × n , where a j l = ∑ π s − 1 ( j , l ) a j π s − 1 ( j , l ) , π s − 1 ( j , l ) is an arbitrary arrangement of s − 1 letters j and m − s letters l , 1 ≤ s ≤ m − 1 , a j j = a j … j , j , l ∈ ⟨ n ⟩ . If there exists a s and 1 ≤ s ≤ m − 1 such that F s ( A ) ∈ R + n × n is irreducible, then A is a weakly essentially irreducible nonnegative tensor.

Remark 2.1

When s = 1 , obviously, F 1 ( A ) = A ˆ , that is, A ˆ is the majorization matrix of A .

In 2013, the concept of diagonal similarity transformation of the tensor was explored and the properties of eigenvalues were introduced in [19].

Definition 2.7

[19] Let A = ( a j 1 j 2 … j m ) ∈ R [ m , n ] , ℬ = ( b j 1 j 2 … j m ) ∈ R [ m , n ] . Then, A and ℬ are said to be diagonally similar, if there exists some invertible diagonal matrix D = diag ( d 1 , d 2 , … , d n ) of order n such that ℬ = A × 1 D − ( m − 1 ) × 2 D × 3 … × m D , where b j 1 j 2 … j m = a j 1 j 2 … j m d j 1 − ( m − 1 ) d j 2 … d j m .

Theorem 2.4

[19] If the two mth order n-dimensional tensors A and ℬ are diagonally similar, then spec ( A ) = spec ( ℬ ) .

3 The proposed algorithm

In this section, we propose an algorithm utilizing displacement transform to find the H -spectral radius of a nonnegative tensor. First, let us review the algorithms presented in [6] and [8].

In 2009, Ng et al. [6] proposed the NQZ method for the largest H -eigenvalue of a nonnegative irreducible tensor.

Algorithm 1 (NQZ Algorithm)
Step 0. Choose x ( 0 ) > 0 , x ( 0 ) ∈ R n . Let y ( 0 ) = A ( x ( 0 ) ) m − 1 and set k ≔ 0 .
Step 1. Compute
x ( k + 1 ) = ( y ( k ) ) 1 m − 1 ‖ ( y ( k ) ) 1 m − 1 ‖ , y ( k + 1 ) = A ( x ( k + 1 ) ) m − 1 ,
λ ̲ ( k + 1 ) = min x j ( k + 1 ) > 0 ( y ( k + 1 ) ) j ( x j ( k + 1 ) ) m − 1 , λ ¯ ( k + 1 ) = max x j ( k + 1 ) > 0 ( y ( k + 1 ) ) j ( x j ( k + 1 ) ) m − 1 .
Step 2. If λ ¯ ( k + 1 ) = λ ̲ ( k + 1 ) , stop. Otherwise, replace k by k + 1 and go to Step 1.

Subsequently, the convergence of the NQZ method was proved for primitive tensors in [12] and for a class of weakly primitive tensors in [17]. Furthermore, it was proved in [9] that the NQZ method exhibits an explicit linear convergence rate for essentially positive tensors. However, it is illustrated in [6] that it does not converge for some irreducible nonnegative tensors.

Example 3.1

Let A = ( a i j k ) ∈ R + [ 3,2 ] , where a 122 = 1 , a 211 = 2 the rest of the elements are 0. Then, for any initial vector x ( 0 ) ∈ R + + 2 , the NQZ Algorithm does not converge.

In 2010, Liu et al. [8] modified the NQZ method and proposed the LZI method.

Algorithm 2 (LZI Algorithm)
Step 0. Choose x ( 0 ) > 0 , x ( 0 ) ∈ R n , γ > 0 . Let ℬ = A + γ ℐ , and set k ≔ 0 .
Step 1. Compute
y ( k ) = ℬ ( x ( k ) ) m − 1 ,
λ ̲ ( k ) = min x j ( k ) > 0 ( y ( k ) ) j ( x j ( k ) ) m − 1 , λ ¯ ( k ) = max x j ( k ) > 0 ( y ( k ) ) j ( x j ( k ) ) m − 1 .
Step 2. If λ ¯ ( k ) = λ ̲ ( k ) , then let λ = λ ¯ ( k ) − γ and stop. Otherwise, compute
x ( k + 1 ) = ( y ( k ) ) 1 m − 1 ‖ ( y ( k ) ) 1 m − 1 ‖ ,
replace k by k + 1 and go to Step 1.

Liu et al. [8] showed that the LZI Algorithm is convergent for a class of irreducible nonnegative tensors. Specifically, when A ∈ R + [ m , n ] is irreducible, ℬ = A + γ ℐ becomes a specially primitive matrix [12], that is, the primitive matrix with b j … j > 0 ( j ∈ ⟨ n ⟩ ) . Therefore, the NQZ Algorithm converges in this case. In 2012, Zhang et al. [10] proved the linear convergence of the LZI Algorithm for weakly positive tensors. Since then, numerous studies have focused on the calculation of the largest eigenvalue of nonnegative tensors. For example, Sheng et al. [20], Yang and Ni [21] and Liu et al. [22] proposed a nonlinear algorithm for the H -spectral radius of nonnegative tensors.

Building upon previous advancements, Zhang and Bu [11] gave a diagonally similar iterative algorithm for calculating the largest H -eigenvalue of a specific class of generalized weakly positive tensors.

Algorithm 3
Step 0. Given A ( 0 ) = A = ( a i 1 i 2 … i m ) , ε − min i ∈ ⟨ n ⟩ a i … i < Δ < r ¯ ( A ( 0 ) ) − min i ∈ ⟨ n ⟩ a i … i , ε > 0 . Set k ≔ 0 .
Step 1. Compute
r i ( A ( k ) ) = ∑ i 2 , … , i m = 1 n a i i 2 … i m ( k ) , i ∈ ⟨ n ⟩ ,
r ¯ ( A ( k ) ) = max i ∈ ⟨ n ⟩ r i ( A ( k ) ) , r ̲ ( A ( k ) ) = min i ∈ ⟨ n ⟩ r i ( A ( k ) ) .
Step 2. If r ¯ ( A ( k ) ) = r ̲ ( A ( k ) ) , then ρ ( A ) = r ¯ ( A ( k ) ) and stop; Otherwise:
Step 3. Set
D ( k ) = ( diag ( r 1 ( A ( k ) ) + Δ , r 2 ( A ( k ) ) + Δ , … , r n ( A ( k ) ) + Δ ) ) 1 m − 1 ,
A ( k + 1 ) = A ( k ) × 1 ( D ( k ) ) − ( m − 1 ) × 2 D ( k ) × 3 … × m D ( k ) ,
and replace k by k + 1 , go to Step 1.

In this work, we continue to study an algorithm with a displacement transformation to compute the H -spectral radius of a nonnegative tensor based on the foundation laid in the literature [18]. We prove the convergence of the algorithm for calculating the H -spectral radius of a weakly irreducible quasi-positive tensor, and give more general conditions for the linear convergence of the algorithm.

Next we present the construction procedure of the proposed algorithm. From the given tensor A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] , we define the sequence of tensors { A ( k ) } k = 0 ∞ , the sequence of vectors { x ( k ) } k = 0 ∞ , and the sequences of numbers { λ ̲ ( k ) } k = 0 ∞ , { λ ¯ ( k ) } k = 0 ∞ as follows:

A ( 0 ) = A = ( a j 1 j 2 … j m ( 0 ) ) , A ( k + 1 ) = A ( k ) × 1 D k − ( m − 1 ) × 2 D k × 3 … × m D k = ( a j 1 j 2 … j m ( k + 1 ) ) , D k = diag ( λ 1 ( k ) , λ 2 ( k ) , … , λ n ( k ) ) + θ k I max j ∈ ⟨ n ⟩ λ j ( k ) + θ k 1 m − 1 ≔ diag ( d 1 ( k ) , d 2 ( k ) , … , d n ( k ) ) ∈ D + n , x ( k ) = ∏ t = 0 k D t e , λ ̲ ( k ) = min j ∈ ⟨ n ⟩ { λ j ( k ) } , λ ¯ ( k ) = max j ∈ ⟨ n ⟩ { λ j ( k ) } , λ j ( k ) = ∑ j 2 , … , j m = 1 n a j j 2 … j m ( k ) ,

where θ k ∈ R , − min j ∈ ⟨ n ⟩ a j … j + a ̲ n ( λ ¯ ( 0 ) ) n − 1 ≤ θ k ≤ λ ¯ ( 0 ) − min j ∈ ⟨ n ⟩ a j … j , a ̲ = min { a j 1 j 2 … j m > 0 : j 1 , j 2 , … , j m ∈ ⟨ n ⟩ } , I ∈ R n × n is unit matrix, e = ( 1, 1 , … , 1 ) T ∈ R + + n . Therefore, there is a ̲ m ( λ ¯ ( 0 ) ) m − 1 ≤ λ j ( k ) + θ k ≤ 2 λ ¯ ( 0 ) − min j ∈ ⟨ n ⟩ a j … j ≕ λ ˜ .

For any x = ( x 1 , x 2 , … , x n ) T , y = ( y 1 , y 2 , … , y n ) T ∈ R + + n , denote x ∘ y = ( x i y i ) ∈ R + + n , y x = ( y j x j ) ∈ R + + n , max { x } = max j ∈ ⟨ n ⟩ { x j } , min { x } = min j ∈ ⟨ n ⟩ { x j } .

From the above definitions, we give the following algorithm to find the spectral radius of the nonnegative tensor.

Algorithm 4
Step 0. Choose x ( 0 ) = ( 1, 1 , … , 1 ) T ∈ R + + n , ε > 0 , θ k ∈ R , let k = 0 .
Step 1. Compute
λ ( k + 1 ) = A ( x ( k ) ) m − 1 ( x ( k ) ) [ m − 1 ] ,
λ ¯ ( k + 1 ) = max { λ ( k + 1 ) } , λ ̲ ( k + 1 ) = min { λ ( k + 1 ) } ,
y ( k + 1 ) = x ( k ) ∘ ( λ ( k + 1 ) + θ k + 1 e ) [ 1 m − 1 ] , x ( k + 1 ) = y ( k + 1 ) ‖ y ( k + 1 ) ‖ ∞ .
Step 2. If λ ¯ ( k + 1 ) − λ ̲ ( k + 1 ) < ε , stop, ρ ( A ) = ( λ ¯ ( k + 1 ) + λ ̲ ( k + 1 ) ) ⁄ 2 . Otherwise, k ≔ k + 1 and go to Step 1.

Remark 3.1

Since the variable parameter we introduce satisfies the conditions at each iteration in Algorithm 4, we call Algorithm 4 an algorithm with displacement transformations to find the H -spectral radius of nonnegative tensors.

Remark 3.2

Let A be an m th-order n -dimensional positive tensor. The computational complexity of Algorithms 1–4 in a single iteration is summarized in Table 1. Specifically, the table lists the number of multiplication/division operations, addition/subtraction operations, and the total number of arithmetic operations required per iteration for each algorithm.

Although Algorithms 1–4 share the same computational complexity of O ( n m ) , the actual number of arithmetic operations per iteration for Algorithm 4 is comparable to that of the NQZ and LZI algorithms. In contrast, Algorithm 3 requires significantly more arithmetic operations, resulting in a higher computational burden in practice.

Table 1

Arithmetic complexity per iteration for four different algorithms

Algorithm	Multiplications/divisions	Additions/subtractions	Total operations
NQZ Algorithm	( m − 1 ) n m + n	n m	m n m + n
LZI Algorithm	( m − 1 ) n m + n	n m	m n m + n
Algorithm 3	n m	n m	( m + 1 ) n m
Algorithm 4	( m − 1 ) n m + 2 n	n m	m n m + 2 n

4 Main results

In this section, we give the definition of a weakly irreducible quasi-positive tensor and prove the convergence of Algorithm 4 to a weakly irreducible quasi-positive tensor. Furthermore, we present a broader condition for the linear convergence of Algorithm 4.

From Algorithm 4 and its construction process, we have the following results:

(4.1) λ j ( k + 1 ) = ∑ j 2 , … , j m = 1 n a j j 2 … j m ( k + 1 ) = ( d j ( k ) ) − 1 ⋅ ∑ j 2 , … , j m = 1 n a j j 2 … j m ( k ) ( d j 2 ( k ) … d j m ( k ) ) 1 m − 1 = … = ∏ t = 0 k d j ( t ) − 1 ⋅ ∑ j 2 , … , j m = 1 n a j j 2 … j m ( 0 ) ∏ t = 0 k ( d j 2 ( t ) … d j m ( t ) ) 1 m − 1 = ∏ t = 0 k d j ( t ) − 1 ⋅ ∑ j 2 , … , j m = 1 n a j j 2 … j m ( 0 ) ∏ t = 0 k d j 2 ( t ) 1 m − 1 … ∏ t = 0 k d j m ( t ) 1 m − 1 = ( ( x ( k ) ) [ 1 − m ] ∘ A ( x ( k ) ) m − 1 ) j = ( x j ( k ) ) 1 − m ⋅ ( A ( x ( k ) ) m − 1 ) j .

For convenience, we introduce the following notation. For the tensor A ( k ) + θ k ℐ ∈ R + [ m , n ] , denote A ( k ) ( θ k ) ≔ A ( k ) + θ k ℐ = ( a j 1 j 2 … j m ( k ) ( θ k ) ) j 1 , … , j m = 1 n , k = 0 , 1 , 2 , … . Let π s ( j 2 , … , j m ) be an arrangement of j 2 , … , j m ∈ ⟨ n ⟩ , π ¯ s = { i s , j 2 , … , j m } \ { i s } = { j 2 , … , j m } ( s = 1 , 2 , … , r ) , and { π ¯ 1 , … , π ¯ r } [ i s ] represent the number of i s ( s = 1,2 , … , m ) in { π ¯ 1 , … , π ¯ r } . To give a definition of the weakly irreducible proposed positive tensor, we first define the simple directed cyclic path generated by the isometrical elements of G ( A ) .

Definition 4.1

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] . For i 1 ∈ ⟨ n ⟩ , if a i 1 π 1 ( j 2 , … , j m ) > 0 , … , a i r π r ( j 2 , … , j m ) > 0 exist, where j 2 , … , j m ∈ γ ¯ = { i 1 , … , i r } , i 1 , … , i r are different from each other and there are i s + 1 ∈ π ¯ s ( s = 1 , 2 , … , r − 1 ) , i r + 1 = i 1 ∈ π ¯ 1 and { π ¯ 1 , … , π ¯ r } [ i s ] = m − 1 , s = 1 , 2 , … , r , we call γ : i 1 → i 2 → … → i r → i 1 the simple directed cyclic path of G ( A ) generated by isometric elements a i 1 π 1 ( j 2 , … , j m ) , … , a i r π r ( j 2 , … , j m ) , and denote ∣ γ ∣ = r as the length of the simple directed cyclic path γ , also known as a i 1 π 1 ( j 2 , … , j m ) , … , a i r π r ( j 2 , … , j m ) on the simple directed cyclic path γ of G ( A ) , and denote a i 1 π 1 ( j 2 , … , j m ) , … , a i r π r ( j 2 , … , j m ) ∈ A [ γ ] .

Let A = ( a j 1 j 2 … j m ) ∈ R [ m , n ] , use C ˆ ( A ) to represent the set of all simple directed cyclic paths generated by the isometrical elements of G ( A ) , and for γ : i 1 → i 2 → … → i r → i 1 ∈ C ˆ ( A ) , use a i 1 π 1 ( j 2 , … , j m ) [ γ ] , … , a i r π r ( j 2 , … , j m ) [ γ ] to represent the elements that generate γ ∈ C ˆ ( A ) .

Let C ∘ ( A ) ⊆ C ˆ ( A ) , denote

( A [ C ∘ ] ) j 1 j 2 … j m = a j 1 j 2 … j m [ γ ] = a j 1 j 2 … j m , γ ∈ C ∘ ( A ) , 0 , otherwise .

Applying simple directed cyclic paths generated by the isometrical elements of G ( A ) , we give the definition of a weakly irreducible quasi-positive tensor.

Definition 4.2

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] . If there exists C ∘ ( A ) ⊆ C ˆ ( A ) that makes G ( A [ C ∘ ] ) irreducible, and for any γ ∈ C ∘ ( A ) , γ is a simple directed cyclic path generated by the isometric elements of G ( A ) , then A is said to be a weakly irreducible quasi-positive tensor.

Below we give an example of a weakly irreducible quasi-positive tensor.

Example 4.1

Let A = ( a j 1 j 2 … j m ) ∈ R + [ 6,7 ] , where a 122335 = 1 , a 222355 = 2 , a 323355 = 2 , a 511111 = 1 , a 466777 = 1 , a 666477 = 2 , a 764444 = 1 , a 333444 = 1 , a 444333 = 2 , the rest of the elements are arbitrary and nonnegative. Denote γ 1 : 1 → 2 → 3 → 5 → 1 , γ 2 : 4 → 6 → 7 → 4 , γ 3 : 6 → 7 → 6 , C ∘ ( A ) = { γ 1 , γ 2 , γ 3 } . Then,

G ( A [ C ∘ ] ) = 0 2 2 0 1 0 0 0 4 2 0 4 0 0 0 2 6 3 4 0 0 0 0 6 4 0 2 3 5 0 0 0 0 0 0 0 0 0 2 0 4 4 0 0 0 4 0 1 0 .

Obviously, G ( A [ C ∘ ] ) is an irreducible matrix, so A is a weakly irreducible quasi-positive tensor.

Remark 4.1

It is evident that essential positive tensors, weakly positive tensors, generalized weakly positive tensors, and weakly essential irreducible nonnegative tensor classes all belong to the category of weakly irreducible quasi-positive tensors.

Remark 4.2

If the optimization matrix A ˆ of A is irreducible, then it is clear that A is a weakly irreducible quasi-positive tensor.

As noted in Remark 4.1 and Remark 4.2, weakly irreducible quasi-positive tensor classes are important extensions of essential positive tensors, weakly positive tensors, generalized weakly positive tensors, weakly essential irreducible nonnegative tensor classes, etc.

By Definitions 2.2 and 4.2, we obtain the following conclusion.

Lemma 4.1

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] be a weakly irreducible quasi-positive tensor. Then, there exists C ∘ ( A ) ⊆ C ˆ ( A ) such that G ( A [ C ∘ ] ) is strongly connected.

In summary, the inclusion relationship among weakly irreducible quasi-positive tensor, weakly essentially irreducible tensor, generalized weakly positive tensor, weakly positive tensor, essentially positive tensor, irreducible nonnegative tensor, and primitive sets of tensors is shown in Figure 1.

$Figure 1 Relations among seven classes of nonnegative tensors.$

Figure 1

Relations among seven classes of nonnegative tensors.

In view of the properties of the weakly irreducible proposed positive tensor yields the following fundamental result.

Lemma 4.2

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] . If A is a weakly irreducible quasi-positive tensor, i.e., there exists C ∘ ( A ) ⊆ C ˆ ( A ) , for any γ ∈ C ∘ ( A ) , γ is a simple directed cyclic path generated by the isometric elements of G ( A ) , then

for any γ ∈ C ∘ ( A ) , we have
∏ s = 1 r a i s π s ( j 2 , … , j m ) [ γ ] ( k ) = ∏ s = 1 r a i s π s ( j 2 , … , j m ) [ γ ] , k ∈ N ;
there exist λ ¯ , λ ̲ > 0 , such that λ ¯ ( k ) ↘ λ ¯ , λ ̲ ( k ) ↗ λ ̲ , and λ ̲ ≤ ρ ( A ) ≤ λ ¯ .
for any a j 1 j 2 … j m [ γ ] = a j 1 j 2 … j m > 0 , γ ∈ C ∘ ( A ) , there exists a > 0 such that a j 1 j 2 … j m [ γ ] ( k ) > a > 0 , k ∈ Z + ∪ { 0 } .

Proof

(1) For any γ ∈ C ∘ ( A ) , denote a i 1 π 1 ( j 2 , … , j m ) [ γ ] ( 0 ) , … , a i r π r ( j 2 , … , j m ) [ γ ] ( 0 ) ∈ A [ γ ] . Since γ is a simple directed cyclic path generated by isometric elements on G ( A ) , it can be inferred from Definition 4.1 that { π ¯ 1 , … , π ¯ r } [ j s ] = m − 1 , s = 1 , 2 , … , r . Therefore, the construction process of the algorithm yields

∏ s = 1 r a i s π s ( j 2 , … , j m ) [ γ ] ( k ) = ∏ s = 1 r a i s π s ( j 2 , … , j m ) [ γ ] ( k − 1 ) ⋅ ∏ s = 1 r ∏ l = 2 m r j l ( k − 1 ) ( θ k − 1 ) ( r i s ( k − 1 ) ( θ k − 1 ) ) m − 1 = ∏ s = 1 r a i s π s ( j 2 , … , j m ) [ γ ] ( k − 1 ) = … = ∏ s = 1 r a i s π s ( j 2 , … , j m ) [ γ ] ( 0 ) .

(2) The proof can be found in [18].

(3) According to the construction of the algorithm, for any γ ∈ C ∘ ( A ) , ∣ γ ∣ = r , we have 0 < a i s π s ( j 2 , … , j m ) ∈ A [ γ ] ( s = 1 , 2 , … , r ) , and then for any k ∈ Z + ,

(4.2) a i s π s ( j 2 , … , j m ) [ γ ] ( k ) = a i s π s ( j 2 , … , j m ) [ γ ] ( k − 1 ) ⋅ ( ∏ l = 2 m r j l ( k − 1 ) ( θ k ) ) 1 m − 1 r i s ( k − 1 ) ( θ k ) = … = a i s π s ( j 2 , … , j m ) [ γ ] ( 0 ) ⋅ ∏ t = 0 k − 1 ( ∏ l = 2 m r j l ( t ) ( θ k ) ) 1 m − 1 r i s ( t ) ( θ k ) ≥ a ̲ ⋅ ∏ t = 0 k − 1 ( ∏ l = 2 m r j l ( t ) ( θ k ) ) 1 m − 1 r i s ( t ) ( θ k ) ,

where a ̲ = min { a j 1 j 2 … j m [ γ ] > 0 : γ ∈ C ∘ ( A ) } .

As can be seen from (4.2)

(4.3) ∏ t = 0 k − 1 ∏ l = 2 m r j l ( t ) ( θ k ) 1 m − 1 r i s ( t ) ( θ k ) ≤ λ ¯ ( 0 ) a ̲ .

It can also be inferred from { π ¯ 1 , … , π ¯ r } [ j s ] = m − 1 , s = 1 , 2 , … , r , that

∏ s = 1 r ∏ t = 0 k − 1 ∏ l = 2 m r j l ( t ) ( θ k ) 1 m − 1 r i s ( t ) ( θ k ) = 1 ,

from equation (4.3), it can be concluded that

∏ t = 0 k − 1 ∏ l = 2 m r j l ( t ) ( θ k ) 1 m − 1 r i s ( t ) ( θ k ) ⋅ λ ¯ ( 0 ) a ̲ r − 1 ≥ 1 ,

∏ t = 0 k − 1 ∏ l = 2 m r j l ( t ) ( θ k ) 1 m − 1 r i s ( t ) ( θ k ) ≥ a ̲ λ ¯ ( 0 ) r − 1 .

Denote r 0 = max { ∣ γ ∣ : γ ∈ C ∘ ( A ) } , then it can be obtained from equation (4.2) that

a i s π s ( j 2 , … , j m ) [ γ ] ( k ) ≥ a ̲ λ ¯ ( 0 ) r − 1 ⋅ a ̲ ≥ a ̲ λ ¯ ( 0 ) r 0 − 1 ⋅ a ̲ ≥ a ̲ λ ¯ ( 0 ) n .

If a = a ̲ λ ¯ ( 0 ) n is defined, then a i s π s ( j 2 , … , j m ) [ γ ] ( k ) ≥ a > 0 , s = 1 , 2 , … , r , ∀ γ ∈ C ∘ ( A ) . □

Let γ j → i represent the simple directed path of G ( A [ C ∘ ] ) from node j to node i , and r j → i = ∣ γ j → i ∣ represent the length of the directed path γ j → i , and r j → i min ( G ( A [ C ∘ ] ) ) = min j ∈ ⟨ n ⟩ \ { i } r j → i , l 0 = max i ∈ ⟨ n ⟩ max j ∈ ⟨ n ⟩ r j → i min ( G ( A [ C ∘ ] ) ) .

Applying Lemma 4.2, we give the convergence of Algorithm 4 to compute the H -spectral radius of a weakly irreducible quasi-positive tensor.

Theorem 4.1

If A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] is a weakly irreducible quasi-positive tensor, then ρ ( A ) = λ ¯ = λ ̲ .

Proof

Since A is a weakly irreducible quasi-positive tensor, it can be inferred from Definition 4.2 that C ∘ ( A ) ⊆ C ˆ ( A ) makes G ( A [ C ∘ ] ) irreducible, and for any γ ∈ C ∘ ( A ) , γ is a simple directed cyclic path generated by the isometric elements of G ( A ) .

(1) For i 0 ∈ { i ∈ ⟨ n ⟩ : λ i ( 0 ) = max i ∈ ⟨ n ⟩ λ i ( 0 ) } , it can be obtained that

λ i 0 ( 1 ) = ∑ j 2 , … , j m = 1 n a i 0 j 2 … j m ( 1 ) = λ ̲ ( 0 ) + ∑ j 2 , … , j m = 1 n a i 0 j 2 … j m ( 0 ) ( θ 0 ) ⋅ ∏ l = 2 m ( r j l ( 0 ) ( θ 0 ) ) 1 m − 1 − ( λ ̲ ( 0 ) ( θ 0 ) ) λ i 0 ( 0 ) ( θ 0 ) ≥ λ ̲ ( 0 ) + a i 0 … i 0 ( 0 ) ( θ 0 ) ⋅ λ ¯ ( 0 ) − λ ̲ ( 0 ) λ i 0 ( 0 ) ( θ 0 ) ≥ λ ̲ ( 0 ) + a λ ˜ ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) .

Similarly, applying the above equation yields

λ i 0 ( t ) ≥ λ ̲ ( 0 ) + a λ ˜ t ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) ( t = 2 , … , l 0 ) .

(2) For i ≠ i 0 , i ∈ ⟨ n ⟩ , because G ( A [ C ∘ ] ) is irreducible, again, in view of Definition 2.3 yields A [ C ∘ ] is a weakly irreducible nonnegative tensor. Hence, by Lemma 4.1, G ( A [ C ∘ ] ) is strongly connected. There are a i 1 j 2 … i 0 … j m [ γ 1 ] > 0 , … , a i r j 2 … i r − 1 … j m [ γ r ] > 0 , i = i r and γ l ∈ C ∘ ( A ) , l = 1 , 2 , … , r ≤ l 0 that make a i l j 2 … i l − 1 … j m ∈ A [ γ l ] . Therefore, applying (3) from Lemma 4.2 and (1) from the proof of Theorem 4.1 yields

λ i 1 ( 2 ) ≥ λ ̲ ( 0 ) + a i 1 j 2 … i 0 … j m ( 1 ) ( θ 1 ) ⋅ ∏ l = 2 m ( λ j l ( 1 ) ( θ 1 ) ) 1 m − 1 − λ ̲ ( 0 ) ( θ 1 ) λ i 1 ( 1 ) ( θ 1 ) ≥ λ ̲ ( 0 ) + a λ ˜ ⋅ ( λ ̲ ( 0 ) ( θ 1 ) ) m − 2 m − 1 ⋅ ( λ i 0 ( 1 ) ( θ 1 ) ) 1 m − 1 − ( λ ̲ ( 0 ) ( θ 1 ) ) 1 m − 1 ≥ λ ̲ ( 0 ) + a λ ˜ ⋅ ( λ ̲ ( 0 ) ( θ 1 ) ) m − 2 m − 1 ⋅ 1 m − 1 ⋅ ( λ i 0 ( 1 ) − λ ̲ ( 0 ) ) ⋅ ( λ ̲ ( 0 ) ( θ 1 ) ) − m − 2 m − 1 ≥ λ ̲ ( 0 ) + a λ ˜ 2 ⋅ 1 m − 1 ⋅ ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) .

Similarly, it obtains that

λ i ( t ) = λ i r ( t ) ≥ λ ̲ ( 0 ) + a λ ˜ t ⋅ 1 ( m − 1 ) t − 1 ⋅ ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) ( t = 3 , … , l 0 ) .

So, for ∀ i ∈ ⟨ n ⟩ , it follows that

λ ¯ ( l 0 ) − λ ̲ ( l 0 ) ≤ λ ¯ ( 0 ) − min i ∈ ⟨ n ⟩ λ i ( l 0 ) ≤ ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) 1 − a λ ˜ l 0 ⋅ 1 ( m − 1 ) l 0 − 1 .

Similar to the proof above, ∀ q ∈ Z + , it has

(4.4) λ ¯ ( q l 0 ) − λ ̲ ( q l 0 ) ≤ ( λ ¯ ( ( q − 1 ) l 0 ) − λ ̲ ( ( q − 1 ) l 0 ) ) 1 − a λ ¯ ( 0 ) + ε l 0 ⋅ 1 ( m − 1 ) l 0 − 1 ≤ … ≤ ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) 1 − a λ ˜ l 0 ⋅ 1 ( m − 1 ) l 0 − 1 q .

If α ≔ 1 − a λ ˜ l 0 ⋅ 1 ( m − 1 ) l 0 − 1 is defined, then 0 < α < 1 , so it obtains that

lim q → ∞ ( λ ¯ ( q l 0 ) − λ ̲ ( q l 0 ) ) ≤ ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) lim q → ∞ α q = 0 .

From (2) in Lemma 4.2, it can be concluded that

λ ¯ − λ ̲ = lim k → ∞ λ ¯ ( k ) − lim k → ∞ λ ̲ ( k ) = lim k → ∞ ( λ ¯ ( k ) − λ ̲ ( k ) ) = 0 .

So, ρ ( A ) = λ ¯ = λ ̲ . □

According to Theorem 4.1, when l 0 = 1 , Algorithm 4 converges linearly. Therefore, Theorem 4.1 reflects the linear convergence of Algorithm 4, which further illustrates the essence of NQZ Algorithm’s linear convergence on essential positive tensors and weak positive tensors.

Regarding the convergence of Algorithm 4, there are some further details as follows.

Theorem 4.2

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] be a weakly irreducible quasi-positive tensor. Then, for Algorithm 4, it has

λ ¯ ( k ) − λ ̲ ( k ) ≤ α k l 0 − 1 ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) , k > l 0 .

Therefore, Algorithm 4 is R -linearly convergent.

Proof

For any k ≥ 1 , there exists q 0 ∈ Z + such that ( q 0 − 1 ) l 0 ≤ k ≤ q 0 l 0 , so q 0 ≥ k l 0 . From equation (4.4), it can be concluded that

λ ¯ ( k ) − λ ̲ ( k ) ≤ α q 0 − 1 ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) ≤ α k l 0 − 1 ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) .

In view of R 1 = limsup k → ∞ ( λ ¯ ( k ) − λ ̲ ( k ) ) 1 k ≤ limsup k → ∞ α k l 0 − 1 ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) 1 k = α 1 l 0 yields that 0 ≤ R 1 < 1 . Therefore, Algorithm 4 is R -linearly convergent.□

By Theorem 4.2, it follows:

Theorem 4.3

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] be a weakly irreducible quasi-positive tensor. For a given 0 < ε < λ ¯ ( 0 ) − λ ̲ ( 0 ) , for Algorithm 4, when k > ln ε − ln ( λ ¯ ( 0 ) − λ ̲ ( 0 ) ) ln α ⋅ l 0 , we must obtain

λ ¯ ( k ) − λ ̲ ( k ) < ε .

Next, we provide the conditions for the linear convergence of Algorithm 4, which utilize the information of the tensor’s directed graph G ( A ) and are broader than the conditions in references.

Theorem 4.4

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] . If A is a weakly irreducible quasi-positive tensor, that is, there exists C ∘ ( A ) ⊆ C ˆ ( A ) to make G ( A [ C ∘ ] ) irreducible, and for any γ ∈ C ∘ ( A ) , γ is a simple directed cyclic path generated by the isometric elements of G ( A ) , and there exists j 0 ∈ ⟨ n ⟩ to make e j j 0 ∈ G ( A [ C ∘ ] ) , j ∈ ⟨ n ⟩ \ { j 0 } , then

λ ¯ ( k + 1 ) − λ ̲ ( k + 1 ) ≤ α ( λ ¯ ( k ) − λ ̲ ( k ) ) ,

where 0 < α < 1 , α = 1 − a λ ˜ .

Proof

Let λ ¯ ( k + 1 ) = λ p ( k + 1 ) , λ ̲ ( k + 1 ) = λ q ( k + 1 ) , denote M = { j 2 … j m : j 2 , … , j m ∈ ⟨ n ⟩ } , and M ( k ) = j 2 … j m : a p j 2 … j m ( k ) ( θ k ) λ p ( k ) ( θ k ) ≥ a q j 2 … j m ( k ) ( θ k ) λ q ( k ) ( θ k ) . Then,

λ ¯ ( k + 1 ) − λ ̲ ( k + 1 ) = λ p ( k + 1 ) − λ q ( k + 1 ) = ∑ j 2 , … , j m = 1 n a p j 2 … j m ( k ) ( θ k ) λ p ( k ) ( θ k ) − a q j 2 … j m ( k ) ( θ k ) λ q ( k ) ( θ k ) ∏ l = 2 m ( λ j l ( k ) ( θ k ) ) 1 m − 1 = ∑ j 2 , … , j m ∈ M ( k ) a p j 2 … j m ( k ) ( θ k ) λ p ( k ) ( θ k ) − a q j 2 … j m ( k ) ( θ k ) λ q ( k ) ( θ k ) ∏ l = 2 m ( λ j l ( k ) ( θ k ) ) 1 m − 1 + ∑ j 2 , … , j m ∈ M \ M ( k ) a p j 2 … j m ( k ) ( θ k ) λ p ( k ) ( θ k ) − a q j 2 … j m ( k ) ( θ k ) λ q ( k ) ( θ k ) ∏ l = 2 m ( λ j l ( k ) ( θ k ) ) 1 m − 1 = ( λ ¯ ( k ) − λ ̲ ( k ) ) ⋅ 1 − ∑ j 2 , … , j m ∈ M \ M ( k ) a p j 2 … j m ( k ) ( θ k ) λ p ( k ) ( θ k ) + ∑ j 2 , … , j m ∈ M ( k ) a q j 2 … j m ( k ) ( θ k ) λ q ( k ) ( θ k ) ≤ ( λ ¯ ( k ) − λ ̲ ( k ) ) 1 − ∑ j 2 , … , j m ∈ M \ M ( k ) a p j 2 … j m [ γ ] ( k ) ( θ k ) + ∑ j 2 , … , j m ∈ M ( k ) a q j 2 … j m [ γ ] ( k ) ( θ k ) λ ˜ ≕ ( λ ¯ ( k ) − λ ̲ ( k ) ) 1 − Δ 1 + Δ 2 λ ˜ .

According to A [ C ∘ ] being weakly irreducible, ( G ( A [ C ∘ ] ) ) j j 0 ≠ 0 , ∀ j ≠ j 0 , j ∈ ⟨ n ⟩ , and Lemma 4.2, for ∀ j ≠ j 0 , there exist ( A [ C ∘ ] ( k ) ( θ k ) ) j j 2 … j 0 … j m = ( A ( k ) ) j j 2 … j 0 … j m > a for δ j j 2 … j 0 … j m = 0 , and ( A [ C ∘ ] ( k ) ) j 0 … j 0 = ( A ( k ) ) j 0 … j 0 + θ k = a j 0 … j 0 + θ k > a .

If p ≠ j 0 , q ≠ j 0 , then a p j 2 … j m [ γ ] ( k ) > a in Δ 1 or a q j 2 … j m [ γ ] ( k ) > a in Δ 2 .
If p = j 0 , then a p … p ( k ) = a j 0 … j 0 + θ k > a in Δ 1 or a q … q ( k ) = a q … q ( 0 ) > a in Δ 2 .
If q = j 0 , then a q … q ( k ) ( θ k ) = a j 0 … j 0 + θ k > a in Δ 1 or a p … p ( k ) ( θ k ) = a p … p + θ k > a in Δ 2 .

Consequently, we have

Δ 1 + Δ 2 > min { a p j 2 … j m [ γ ] ( k ) , a q j 2 … j m [ γ ] ( k ) , a j 0 … j 0 ( k ) ( θ k ) } ≥ min a ̲ a ̲ λ ¯ ( 0 ) r 0 − 1 , ε = a > 0 .

Therefore,

λ ¯ ( k + 1 ) − λ ̲ ( k + 1 ) ≤ ( λ ¯ ( k ) − λ ̲ ( k ) ) 1 − a λ ˜ ≕ α ( λ ¯ ( k ) − λ ̲ ( k ) ) ,

where α = 1 − a λ ˜ , 0 < α < 1 .□

Corollary 4.1

If A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] is the essentially positive tensor or the weakly positive tensor or the generalized weakly positive tensor, then the Algorithm 4 for calculating their H-spectral radius converges, and there is

λ ¯ ( k + 1 ) − λ ̲ ( k + 1 ) ≤ α ( λ ¯ ( k ) − λ ̲ ( k ) ) ,

where α = 1 − a λ ˜ , 0 < α < 1 .

The following example shows that the convergence condition for Algorithm 4 given in Theorem 4.4 is weaker than that in the literature [9–11,18].

Example 4.2

Let A = ( a i j k ) ∈ R + [ 3,3 ] , where a 112 = 1 , a 233 = 1 , a 312 = 1 , a 211 = 1 , the rest of the elements are 0. Then, A is a weakly irreducible quasi-positive tensors, since

G ( A ) = 1 1 0 2 0 2 1 1 0 .

Applying Algorithm 4 and Theorem 4.4, we have λ ¯ ( k + 1 ) − λ ̲ ( k + 1 ) ≤ α ( λ ¯ ( k ) − λ ̲ ( k ) ) , where 0 < α < 1 , the values of the parameter α are given in Theorem 4.1. However, the majorization matrix A ˆ = 0 for A . By Definition 2.6,

F 2 ( A ) = 0 1 0 1 0 2 0 0 0

is reducible and therefore, does not satisfy the convergence condition in the literature [9–11,18].

Remark 4.3

By Theorem 4.4, a broader condition for the linear convergence of Algorithm 4 for calculating H -spectral radius of a nonnegative tensor is provided, which extends the results of the linear convergence of the algorithm presented in the literature [9–11,18].

5 Numerical examples

Example 5.1

Let A = ( a j 1 j 2 … j m ) ∈ R + [ m , n ] , where a j , j + 1 , j + 2 … j + 2 = j j + 1 , j = 1 , 2 , … , n − 2 , a n − 1 , n , 1 … 1 = n − 1 n , a n , 1 , 2 … 2 = 1 , the rest of the elements are 0.

From Example 5.1, it can be verified that A is a weakly irreducible quasi-positive tensor, as

G ( A ) = 0 1 2 m − 2 2 0 … 0 0 0 0 2 3 2 ( m − 2 ) 3 … 0 0 ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ 0 0 0 0 … n − 2 n − 1 ( m − 2 ) ( n − 2 ) n − 1 ( m − 2 ) ( n − 1 ) n 0 0 0 … 0 n − 1 n 1 m − 2 0 0 … 0 0

is irreducible. Taking ε = 1 0 − 8 , θ k = ( λ ¯ ( k ) + λ ̲ ( k ) ) ⁄ 2 . The H -spectral radius of tensor A was computed using four different methods: Algorithm 4 (proposed in this work), the NQZ Algorithm, the LZI Algorithm, and Algorithm 3. A comparative study in terms of the number of iterations and corresponding CPU times was conducted, with the results summarized in Table 2.

Table 2

Comparison of Algorithm 4, NQZ Algorithm, LZI Algorithm, and Algorithm 3

( m , n )	Algorithm 4		NQZ Algorithm		LZI Algorithm ( γ = 1 )		Algorithm 3 ( Δ = 1 )
	Iter	CPU (s)	Iter	CPU (s)	Iter	CPU (s)	Iter	CPU (s)	ρ ( A )
(3, 5)	28	0.0215	81	0.0307	32	0.0257	32	0.0513	0.7284
(3, 10)	123	0.0421	340	0.0324	127	0.0190	127	0.0317	0.7943
(3, 20)	495	0.1884	1362	0.3707	505	0.1886	505	0.2961	0.8609
(3, 40)	1957	3.5082	5377	7.8504	1978	3.4932	1978	6.6170	0.9119
(3, 60)	4352	23.7475	11954	54.7203	4384	24.1691	4394	48.0390	0.9340

As shown in Table 2, Algorithm 4 exhibits superior overall computational performance among the four methods. When compared to the NQZ Algorithm, Algorithm 4 requires substantially fewer iterations across all tested cases. For example, at ( m , n ) = (3, 60), the iteration count drops from 11,954 (NQZ) to 4,352 (Algorithm 4), representing a reduction of over 63%. In terms of CPU time, the advantage is even more pronounced in high-dimensional settings: Algorithm 4 completes in 23.75 s, less than half the time required by the NQZ Algorithm (54.72 s). Compared to the LZI Algorithm, Algorithm 4 demonstrates slightly better performance. Although the number of iterations is similar in both methods, Algorithm 4 consistently achieves marginally faster convergence. In particular, its CPU time remains slightly lower or nearly identical across all test cases, while maintaining fewer iteration steps. When contrasted with Algorithm 3, Algorithm 4 shows a significant advantage in computational efficiency. Although Algorithm 3 shares the same number of iterations as the LZI Algorithm, its CPU time is consistently the highest among all methods. This indicates that its per-iteration computational cost is much higher, making it less suitable for large-scale problems.

Example 5.2

Consider a random tensor A ∈ R + [ m , n ] , whose all entries of random values are drawn from the standard uniform distribution on (0, 1).

From Example 5.2, it is clear that A is a positive tensor. Taking ε = 1 0 − 8 , θ k = − min { a i i i i } + ε . The H -spectral radius of tensor A was computed using four different methods: Algorithm 4 (proposed in this study), the NQZ Algorithm, the LZI Algorithm, and Algorithm 3. The number of iterations and corresponding CPU times for each method are reported in Table 3.

Table 3

Comparison of Algorithm 4, NQZ Algorithm, LZI Algorithm, and Algorithm 3

( m , n )	Algorithm 4		NQZ Algorithm		LZI Algorithm ( γ = 1 )		Algorithm 3 ( Δ = 1 )
	Iter	CPU (s)	Iter	CPU (s)	Iter	CPU (s)	Iter	CPU (s)	ρ ( A )
(4, 10)	5	0.2149	4	0.0431	5	0.0212	5	0.0888	499.4330
(4, 20)	4	0.2399	3	0.0702	4	0.0474	4	0.3069	4.0004 × 1 0 3
(4, 40)	4	1.0637	3	0.8860	4	0.8664	4	4.5666	3.2017 × 1 0 4
(4, 60)	4	4.9514	3	4.5107	4	4.6331	4	23.6872	1.0797 × 1 0 5
(4, 80)	4	15.9150	3	15.0882	4	16.2588	4	77.1908	2.5597 × 1 0 5

As shown in Table 3, Algorithm 4 exhibits performance comparable to that of the LZI Algorithm in all dimensions tested. Specifically, both methods require the same number of iterations in most cases, and their computation times are nearly identical. Compared to the NQZ Algorithm, Algorithm 4 generally requires one more iteration but shows a slightly longer computation time. This is mainly due to the more complex per-iteration operations in Algorithm 4, despite its faster convergence in other settings. In contrast, Algorithm 3 consistently incurs the highest computational cost among all four methods. Although it shares the same number of iterations with the LZI Algorithm and Algorithm 4, its CPU time is significantly larger, indicating a much higher per-iteration overhead. For example, at ( m , n ) = (4, 80), Algorithm 3 takes 77.19 s, while Algorithm 4 only takes 15.92 s.

In summary, the results of Examples 5.1 and 5.2 collectively demonstrate that Algorithm 4 offers a balanced and robust performance. Compared to the NQZ Algorithm, it achieves better scalability in high-dimensional settings. Compared to the LZI Algorithm, it achieves similar iteration counts with slightly increased cost. Compared to Algorithm 3, it provides significantly reduced computation time. These findings validate the effectiveness and practical applicability of the proposed Algorithm 4 in computing the H -spectral radius of both structured and randomly generated nonnegative tensors.

6 Conclusion

In this article, a class of weakly irreducible quasi-positional tensors is defined, and an algorithm is given to compute their H -spectral radius. It is proved that the algorithm is R -linearly convergent, and the general conditions for linear convergence of the algorithm are given. Numerical examples show that Algorithm 4 has priority over NQZ Algorithm, LZI Algorithm, and Algorithm 3.

Acknowledgments

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

Funding information: This work was supported by the National Natural Science Foundation of China [No. 61772292] and the Fujian Provincial Natural Science Foundation of China [Nos 2023J01996, 2023J01997, 2024J01874].
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and have consented to its submission to the journal, reviewed all results, and approved the final version of the manuscript. Both authors contributed equally to this work.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: No data sets were generated or analyzed during the current study.

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Received: 2025-01-15

Revised: 2025-06-27

Accepted: 2025-08-02

Published Online: 2025-09-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2025-0194

Keywords for this article

quasi-positive tensor; weakly irreducible; H-spectral radius; algorithm; linear convergence

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