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

1.   Introduction

Consider an $m$-order $n$-dimensional square tensor $\mathcal{A}$ consisting of $n^m$ entries in the real field $\mathbb{R}$:

If $a_{i_1i_2 \cdots i_m}\geq 0, 1\le i_1, i_2, \cdots, i_m\le n,$ then $\mathcal{A}$ is called an $m$-order $n$-dimensional nonnegative tensor. Denote the set of all $m$-order $n$-dimensional nonnegative tensors as $\mathbb{R}_+^{[m, n]}$. $\mathbb{R}^n$, $\mathbb{R}^n_{+}$, $\mathbb{R}^n_{++}$ 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 $\lambda$ and a nonzero real vector ${\bf x}\in\mathbb{R}^n$ satisfy the following homogeneous polynomial equation:

where

is an $n$-dimensional vector and

then $\lambda$ is an $H$-eigenalue of $\mathcal{A}$ and ${\bf x}$ is an $H$-eigenector of $\mathcal{A}$ associated with $\lambda$. Define the set of $H$-eigenvalues of $\mathcal{A}\in \mathbb{R}_+^{[m, n]}$ as $\sigma_H(\mathcal{A})$ and $\rho(\mathcal{A}) = \max\limits_{\lambda\in \sigma_H(\mathcal{A})}|\lambda|$ as the $H$-spectral radius of tensor $\mathcal{A}$. Let $\sigma(\mathcal{A})$ be the set of eigenvalues of a tensor $\mathcal{A}$.

Let $\mathcal{A} = (a_{i_1i_2 \cdots i_m})\in \mathbb{R}_+^{[m, n]}$. 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 $\mathcal{E} = (\delta_{i_1i_2\cdots i_m})$ be the $m$-order $n$-dimensional unit tensor whose entries are

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.

2.   Preliminaries

In this section, we mainly introduce some related concepts and important properties of tensors and matrices. For a positive integer $n$, let $\langle n\rangle = \{1, 2, \ldots, n\}$.

Definition 2.1. ^[12] An $m$-order $n$-dimensional tensor $\mathcal{A}$ is called reducible if there exists a nonempty proper index subset $I\subset\langle n\rangle$ such that

If $\mathcal{A}$ is not reducible, then $\mathcal{A}$ is irreducible.

Definition 2.2. ^[13] A nonnegative matrix $\mathring{\mathcal{A}}$ is called the majorization associated to nonnegative tensor $\mathcal{A}$ if the $(i, j)$-th element of $\mathring{\mathcal{A}}$ is defined to be $a_{ij\cdots j}$ for any $i, j = 1, \cdots, n$.

Definition 2.3. ^[13] A nonnegative $m$-order $n$-dimensional tensor $\mathcal{A}$ is essentially positive if $\mathcal{A}{\bf x}^{m-1}\in \mathbb{R}_{++}^n$ for any nonzero ${\bf x}\in \mathbb{R}_{+}^n$.

Definition 2.4. ^[9] Let $\mathcal{A}$ be a nonnegative tensor of order $m$ and dimension $n$. $\mathcal{A}$ is weakly positive if

Definition 2.5. ^[11] Let $\mathcal{A} = (a_{i_1i_2 \cdots i_m})\in \mathbb{R}_+^{[m, n]}$, then $\mathcal{A}$ is generalized weakly positive if there exists $i_0\in \langle n\rangle$, such that $a_{i_0 j\cdots j} > 0, a_{ji_0 \cdots i_0} > 0$ for all $j\in \langle n\rangle \backslash \{i_0\}$.

Definition 2.6. ^[14] Let $\mathcal{A} = (a_{i_1i_2 \cdots i_m})\in \mathbb{R}_+^{[m, n]}$.

(1) We call a nonnegative matrix $G(\mathcal{A})$ the representation associated to nonnegative tensor $\mathcal{A}$ if the $(i, j)$-th element of $G(\mathcal{A})$ is defined to be the summation of $a_{\{ii_2\cdots i_m\}}$ with indices ${\{i_2\cdots i_m\}}\ni j.$

(2) We call $\mathcal{A}$ weakly reducible if its representation $G(\mathcal{A})$ is a reducible matrix and weakly primitive if $G(\mathcal{A})$ is a primitive matrix. If $\mathcal{A}$ is not weakly reducible, then it is called weakly irreducible.

Definition 2.7. Let $\mathcal{A} = (a_{i_1i_2 \cdots i_m})\in \mathbb{R}_+^{[m, n]}$ and $\pi_{s-1}(i, j)$ be an arrangement of $s-1$ letters $i$ and $m-s$ letters $j$. If there exists $s\in\langle m-1\rangle$ and $i_0\in\langle n\rangle$ for any $j\in\langle n\rangle, j\neq i_0$, such that $a_{i_0 \pi_{s-1}(i_0, j)}\neq 0$ and $a_{j\pi_{s-1}(j, i_0)}\neq 0$ hold, then $\mathcal{A}$ is called an $s$-index weakly positive tensor.

For example, $\mathcal{A} = (a_{ijk})\in\mathbb{R}_+^{[3, 3]},$ where $a_{113} = 1, a_{223} = 4, a_{331} = 2, a_{332} = 5$ and $a_{i_1i_2i_3}\geq 0 (1\le i_1, i_2, i_3\le 3)$ elsewhere, then $\mathcal{A}$ 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 $\mathcal{A} = (a_{i_1i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$, $\rho(\mathcal{A})$ is an eigenvalue with a nonnegative eigenvector ${\bf x}\in \mathbb{R}_{+}^n$ corresponding to it.

Theorem 2.2. ^[16] Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_+^{[m, n]}$. $\rho(\mathcal{A})$ is the spectral radius of $\mathcal{A}$, then

Theorem 2.3. ^[12] If $\mathcal{A}$ is an irreducible nonnegative tensor of order $m$ and dimension $n$, then there exists $\lambda_0 > 0$ and ${\bf x}_0 > 0, {\bf x}_0\in \mathbb{R}^n$ such that $\mathcal{A}{\bf x}_0^{m-1} = \lambda_0 {\bf x}_0^{[m-1]}.$ Moreover, if $\lambda$ is an eigenvalue with a nonnegative eigenvector, then $\lambda = \lambda_0$. If $\lambda$ is an eigenvalue of $\mathcal{A}$, then $|\lambda|\le \lambda_0$.

Definition 2.8. ^[17] Let $\mathcal{A} = (a_{i_1i_2 \cdots i_m})\in \mathbb{R}_+^{[m, n]}$, $\mathcal{B} = (b_{i_1i_2 \cdots i_m})\in \mathbb{R}_+^{[m, n]}$. The tensors $\mathcal{A}$ and $\mathcal{B}$ are said to be diagonal similar if there exists some invertible diagonal matrix $D = \mathrm{diag}(d_{1}, d_{2}, \cdots, d_{n})$ of order $n$ such that $\mathcal{B} = \mathcal{A}\times_{1} D^{-(m-1)}\times_{2} D\times_{3} \cdots \times_{m} D$, where $b_{i_{1}i_{2}\cdots i_{m}} = a_{i_{1}i_{2}\cdots i_{m}}d^{-(m-1)}_{i_{1}}d_{i_{2}}\cdots d_{i_{m}}$.

Theorem 2.4. ^[17] If the two $m$-order $n$-dimensional tensors $\mathcal{A}$ and $\mathcal{B}$ are diagonal similar, then $\sigma(\mathcal{A}) = \sigma(\mathcal{B})$.

3.   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.

Let $\mathcal{A} = \mathcal{A}^{(0)} = (a_{i_1i_2 \cdots i_m}^{(0)})\in \mathbb{R}_+^{[m, n]}$, $r_i(\mathcal{A}^{(0)}) = \sum\limits_{i_{2}, \cdots, i_{m} = 1}^{n}a^{(0)}_{ii_{2}\cdots i_{m}}(i\in\langle n\rangle)$ and $\overline r(\mathcal{A}^{(0)}) = \max\limits_{i\in\langle n \rangle}r_i(\mathcal{A}^{(0)})$. $\varepsilon$ is a sufficiently small positive number. $\alpha_k\in\mathbb{R}(k = 0, 1, 2, \cdots)$ satisfies $\varepsilon-\min\limits_{i\in\langle n\rangle}a_{ii\cdots i} < \alpha_k\le\overline r(\mathcal{A}^{(0)})$.

In the following, we will give the convergence condition of Algorithm 3.

Define $\mathcal{A}^{(k)}+\alpha_k \mathcal{E} = \mathcal{A}^{(k)}(\alpha_k) = :(a_{i_{1}i_{2}\cdots i_{m}}(\alpha_k))_{i_{1}, i_{2}, \cdots, i_{m} = 1}^n$.

Lemma 3.1. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$. For tensor sequence $\mathcal{A}^{(l)}(l = 0, 1, 2, \cdots),$ we have $\overline{r}(\mathcal{A}^{(l)})(l = 0, 1, 2, \cdots)$ as monotonically decreasing with lower bound, and $\underline{r}(\mathcal{A}^{(l)})(l = 0, 1, 2, \cdots)$ as monotonically increasing with upper bound.

Proof. Notice that

then $\overline{r}(\mathcal{A}^{(l+1)})\le \overline{r}(\mathcal{A}^{(l)}), l = 0, 1, 2, \cdots$. Therefore, $\overline{r}(\mathcal{A}^{(l)})$ is monotonically decreasing. Similarly, it can be proved that $\underline{r}(\mathcal{A}^{(l+1)})\geq \underline{r}(\mathcal{A}^{(l)}), l = 0, 1, 2, \cdots$, so $\underline{r}(\mathcal{A}^{(l)})$ is monotonically increasing. By $\underline{r}(\mathcal{A}^{(l)})\le \overline{r}(\mathcal{A}^{(l)})$ we can obtain that $\overline{r}(\mathcal{A}^{(l)})(l = 0, 1, 2, \cdots)$ is monotonically decreasing with lower bound $\underline{r}(\mathcal{A}^{(0)})$, and $\underline{r}(\mathcal{A}^{(l)})(l = 0, 1, 2, \cdots)$ is monotonically increasing with upper bound $\overline{r}(\mathcal{A}^{(0)})$. □

Lemma 3.2. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$ be an $s$-index weakly positive tensor, then it is an irreducible tensor.

Proof. Because $\mathcal{A}$ is an $s$-index weakly positive tensor, by Definition 2.7, there are $s\in\langle m-1\rangle$ and $i_0\in\langle n\rangle$ such that $a_{i_0 \pi_{s-1}(i_0, j)}\neq 0$, $a_{j\pi_{s-1}(j, i_0)}\neq 0$, $j\in\langle n\rangle, j\neq i_0$. That is, for any $j\in I\subset \langle n\rangle$, there is $i_0\in \langle n\rangle \backslash I$ such that $a_{j\pi_{s-1}(j, i_0)}\neq 0$. By Definition 2.1, $\mathcal{A}$ is an irreducible tensor. □

Lemma 3.3. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$ be an $s$-index weakly positive tensor. That is, there are $s\in\langle m-1\rangle$ and $i_0\in\langle n\rangle$ for any $j\in\langle n\rangle, j\neq i_0$, such that $a_{i_0 {\pi_{s-1}(i_0, j)}}\neq 0$ and $a_{j{\pi_{s-1}(j, i_0)}}\neq 0$ hold, then for any $k\in \mathbb{N}$,

Proof. If $a_{j {\pi_{s-1}(j, i_0)}}\neq 0$, then $a_{j {\pi_{s-1}(j, i_0)}}^{(k)}\neq 0, a_{i_0 i_0 \cdots i_0}^{(k)} = a_{i_0 i_0 \cdots i_0}, k = 0, 1, 2, \cdots$. In the case of $j\neq i_0$, it can be obtained from Lemma 3.1 that

where $\tilde{a} = \min\limits_{j\in\langle n \rangle\backslash\{i_0\}}\{a_{i_0 {\pi_{s-1}(i_0, j)}}, a_{j{\pi_{s-1}(j, i_0)}}\}$, then

Then, by

we have

then $a_{i_0 {\pi_{s-1}(i_0, j)}}^{(k)}\geq\frac{\tilde{a}^2}{\overline{r}(\mathcal{A}^{(0)})}(j\neq i_0)$ can be obtained by (3.1).

In the case of $j = i_0$, it holds that $a_{i_0 i_0 \cdots i_0}^{(k)}(\alpha_k) = a_{i_0 i_0 \cdots i_0}^{(k)}+\alpha_k\geq\min\limits_{i\in\langle n \rangle}a_{ii\cdots i}-\min\limits_{i\in\langle n \rangle}a_{ii\cdots i}+\varepsilon = \varepsilon$, then $a_{j{\pi_{s-1}(j, i_0)}}^{(k)}(\alpha_k)\geq \min\{\frac{\tilde{a}^2}{\overline{r}(\mathcal{A}^{(0)})}, \varepsilon \}$. □

Theorem 3.1. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$ be an $s$-index weakly positive tensor. That is, there are $s\in\langle m-1\rangle$ and $i_0\in\langle n\rangle$ for any $j\in\langle n\rangle, j\neq i_0$, such that $a_{i_0 {\pi_{s-1}(i_0, j)}}\neq 0$ and $a_{j{\pi_{s-1}(j, i_0)}}\neq 0$ hold, then for Algorithm 3,

where $\alpha = 1-\frac{\hat{a}^2}{2\overline{r}(\mathcal{A}^{(0)})}, \overline{r}(\mathcal{A}^{(0)}) = \max\limits_{i\in\langle n \rangle}r_i(A), \hat{a} = \min\limits_{j\in\langle n \rangle\backslash\{i_0\}}\{\frac{\tilde{a}^2}{\overline{r}(\mathcal{A}^{(0)})}, \varepsilon\}$, then we can get

Proof. Let $A^{[0]} = (a_{ij}^{[0]})_{n\times n}, a_{i_0 j} = a_{i_0 {\pi_{s-1}(i_0, j)}}, a_{ji_0} = a_{j{\pi_{s-1}(j, i_0)}}, j\in\langle n\rangle$ and zero elsewhere, then $A^{[0]}$ is an irreducible matrix. Let $\overline{r}(\mathcal{A}^{(k+1)}) = r_{p}^{(k+1)}, \underline{r}(\mathcal{A}^{(k+1)}) = r_{q}^{(k+1)}$, then

Denote $I = \{i_2\cdots i_m|i_2, \cdots, i_m\in\langle n\rangle\}$ and

then

Therefore, either $\sum\limits_{i_{2}, \cdots, i_{m}\in I\backslash I(k)}a_{p i_2\cdots i_m}^{(k)}$ includes $a^{(k)}_{p {\pi_{s-1}(p, i_0)}}$ or $\sum\limits_{i_{2}, \cdots, i_{m}\in I(k)}a_{q i_2\cdots i_m}^{(k)}$ includes $a^{(k)}_{q {\pi_{s-1}(q, i_0)}}$, then

Thus,

where $\alpha = 1-\frac{\hat{a}}{2\overline{r}(\mathcal{A}^{(0)})}$. Therefore,

Note that $0 < \alpha = 1-\frac{\hat{a}}{2\overline{r}(\mathcal{A}^{(0)})} < 1$, and we can obtain $\lim\limits_{k\rightarrow \infty}(\overline{r}(\mathcal{A}^{(k)})-\underline{r}(\mathcal{A}^{(k)})) = 0$. From Lemma 3.1,

□

Corollary 3.1. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}, \varepsilon > 0$. If $\mathcal{A}$ is an $s$-index weakly positive tensor, then by Algorithm 3 there must be

that satisfies $\overline{r}(\mathcal{A}^{(K)})-\underline{r}(\mathcal{A}^{(K)}) < \varepsilon$, where $\alpha$ is defined in Theorem 3.1.

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.

Corollary 3.2. If $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$ is an essentially positive tensor or a weakly positive tensor, then

where $\alpha = 1-\frac{\hat{a}}{2\overline{r}(\mathcal{A}^{(0)})}, \; \overline{r}(\mathcal{A}^{(0)}) = \max\limits_{i\in\langle n \rangle}r_i(\mathcal{A}), \hat{a} = \min\{\frac{\tilde{a}^2}{\overline{r}(\mathcal{A}^{(0)})}, \varepsilon\}, \; \tilde{a} = \min\limits_{j\neq i}\{a_{ij\cdots j}\}$ and, thus,

Corollary 3.2 further confirms the linear convergence of the LZI algorithm for weakly essentially positive tensors in Theorem 4.1 in ^[9].

From above, the inclusion relationship is shown among irreducible tensor, primitive tensor, $s$-index weakly positive tensor, generalized weakly positive tensor, weakly positive tensor and essentially positive tensor sets in Figure 1.

Theorem 3.2. Let $\mathcal{A} = (a_{i_{1}i_{2}\cdots i_{m}})\in \mathbb{R}_{+}^{[m, n]}$ be an $s$-index weakly positive tensor. For the positive diagonal matrix $D_i(i = 0, 1, 2, \cdots)$ in the construction process of Algorithm 3, define $D^{(t)} = \prod\limits_{i = 0}^{t}D_i$, then $\lim\limits_{k\rightarrow \infty}D^{(t)}$ exists and denote it as $\hat{D}$. Then, ${\bf x} = \hat{D}{\bf e}\in\mathbb{R}_{++}^{n}$ satisfies $\mathcal{A}{\bf x}^{m-1} = \rho(\mathcal{A}){\bf x}^{[m-1]}$, where ${\bf e} = (1, 1, \cdots, 1)^T\in\mathbb{R}_{++}^{n}$.

Proof. From the construction of Algorithm 3, it is known that the sequence of diagonal elements $(D^{(t)})_{jj}(j = 1, 2, \cdots, n)$ of the positive diagonal matrix $D^{(t)}(j = 1, 2, \cdots, n)$ monotonically decreases with a lower bound, so $\lim\limits_{k\rightarrow \infty}D^{(t)}$ exists. According to Lemma 1, the sequence of each element corresponding to tensor sequence $\{\mathcal{A}^{(k)}\}_{k = 0}^{\infty}$ is nonnegative and has an upper bound, so there is a convergent tensor subsequence of $\{\mathcal{A}^{(k)}\}_{k = 0}^{\infty}$ marked as $\{\mathcal{A}^{(k_l)}\}_{l = 0}^{\infty}$, and denote $\lim\limits_{k\rightarrow \infty}\mathcal{A}^{(k_l)} = \hat{\mathcal{A}}$. Take the limit on both sides of $\mathcal{A}^{(k_{l+1})} = \mathcal{A}\times_1(D^{(k_l)})^{m-1}\times_2 D^{(k_l)}\times_3 \cdots \times_m D^{(k_l)}$ to get

that is,

It can be seen from Theorem 2.4 that $r_i(\hat{\mathcal{A}}) = \rho(\mathcal{A}), i\in\langle n \rangle$; therefore

where ${\bf e} = (1, 1, \cdots, 1)^T\in\mathbb{R}_{++}^{n}$. Denote ${\bf x} = \hat{D}{\bf e}$, then ${\bf x} = \hat{D}{\bf e}\in\mathbb{R}_{++}^{n}$, and $\mathcal{A}{\bf x}^{m-1} = \rho(\mathcal{A}){\bf x}^{[m-1]}$. □

4.   Numerical examples

In this section, to show the effectiveness of Algorithm 3, we compare it with the LZI algorithm. For the parameter $\alpha_k$ in the algorithm, we selected different values and compared the corresponding results.

Example 4.1. Let $\mathcal{A} = (a_{i_1 i_2 i_3 i_4 i_5})\in \mathbb{R}_+^{[m, n]}(m = 5)$, where $a_{111jj} = 1, a_{jjj11} = 1$ for all $j\in \langle n\rangle \backslash \{1\}$ and zero elsewhere.

In the experiment, $\alpha_k = (\overline r(\mathcal{A}^{(k)})-\underline r(\mathcal{A}^{(k)}))/mn$ in Algorithm 3, and ${\bf x}^{(0)} = {\bf e} = (1, 1, \cdots, 1)^T$ $\in \mathbb{R}_{++}^n$, $\rho = 1$ in the LZI algorithm. We terminated the iteration when one of the conditions below was met:

(1) $\overline r(\mathcal{A}^{(k)})-\underline r(\mathcal{A}^{(k)}) < 10^{-8}$.

(2) The number of iteration exceeds $10^{4}$.

Some numerical results are given in Table 1, where $\rho(\mathcal{A})$ denotes the $H$-spectral radius of $\mathcal{A}$, 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.

Example 4.2. Consider a random tensor $\mathcal{A}\in \mathbb{R}_+^{[m, n]}(m = 3)$, 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 $\varepsilon = 10^{-8}$ and the termination conditions are the same as in Example 1. Take different values for $\alpha_k$ in Algorithm 3, and the corresponding results are shown in Table 2, where $\overline r-\underline r = \overline r(\mathcal{A}^{(k)})-\underline r(\mathcal{A}^{(k)})$.

From the data in Table 2, it can be seen that when the value of $\alpha_k$ 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 $\alpha_k$ will improve the efficiency of the algorithm.

5.   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.

Acknowledgments

This work is supported by the Natural Sciences Program of Science and Technology of Jilin Province of China (20190201139JC).

Conflict of interest

The authors declare that there are no conflict of interest.