Infinity norm upper bounds for the inverse of S DD k matrices

: In this paper, we introduce a new subclass of H -matrices called S DD k matrices, which contains S DD matrices and S DD 1 matrices as special cases, and present some properties of S DD k matrices. Based on these properties, we also provide new infinity norm bounds for the inverse of S DD matrices and S DD k matrices. It is proved that these new bounds are better than some existing results in some cases. Numerical examples demonstrate the e ff ectiveness of the obtained results


Introduction
Let n be an integer number, N = {1, 2, . . . , n}, and let C n×n be the set of all complex matrices of order n. A matrix M = (m i j ) ∈ C n×n (n ≥ 2) is called a strictly diagonally dominant (S DD) matrix [1] if It was shown that an S DD matrix is an H-matrix [1], where matrix M = (m i j ) ∈ C n×n (n ≥ 2) is called an H-matrix [1][2][3] if and only if there exists a positive diagonal matrix X such that MX is an S DD matrix [1,2,4]. H-matrices are widely applied in many fields, such as computational mathematics, economics, mathematical physics and dynamical system theory, see [1,[4][5][6]. Meanwhile, upper bounds for the infinity norm of the inverse matrices of H-matrices can be used in the convergence analysis of matrix splitting and matrix multi-splitting iterative methods for solving the large sparse of linear equations [7], as well as linear complementarity problems. Moreover, upper bounds of the infinity norm of the inverse for different classes of matrices have been widely studied, such as CKV-type matrices [8], S -S DDS matrices [9], DZ and DZ-type matrices [10,11], Nekrasov matrices [12][13][14][15], S -Nekrasov matrices [16], Q-Nekrasov matrices [17], GS DD 1 matrices [18] and so on.
In 2011, Peňa [19] proposed a new subclass of H-matrices called S DD 1 matrices, whose definition is listed below. A matrix M = (m i j ) ∈ C n×n (n ≥ 2) is called an S DD 1  Subsequently, some upper bounds for the infinite norm of the inverse matrices of S DD matrices, S DD 1 matrices and GS DD 1 matrices are presented, see [18,20,21]. For example, the following results that will be used later are listed. Theorem 1. (Varah bound) [21] Let matrix M = (m i j ) ∈ C n×n (n ≥ 2) be an S DD matrix. Then .

Theorem 4. [18]
Let M = (m i j ) ∈ C n×n be a GS DD 1 matrix. Then On the basis of the above articles, we continue to study special structured matrices and introduce a new subclass of H-matrices called S DD k matrices, and provide some new upper bounds for the infinite norm of the inverse matrices for S DD matrices and S DD k matrices, which improve the previous results. The remainder of this paper is organized as follows: In Section 2, we propose a new subclass of H-matrices called S DD k matrices, which include S DD matrices and S DD 1 matrices, and derive some properties of S DD k matrices. In Section 3, we present some upper bounds for the infinity norm of the inverse matrices for S DD matrices and S DD k matrices, and provide some comparisons with the well-known Varah bound. Moreover, some numerical examples are given to illustrate the corresponding results.

S DD k matrices
In this section, we propose a new subclass of H-matrices called S DD k matrices, which include S DD matrices and S DD 1 matrices, and derive some new properties.
Immediately, we know that S DD k matrices contain S DD matrices and S DD 1 matrices, so Proof. For ∀i ∈ N 1 (M), from Definition 1, it holds that |m ii | > p (k−1) i (M). For ∀i ∈ N 2 (M), we have that |m ii | > r i (M). When k = 2, it follows that By induction, we obtain that for ∀i ∈ N 2 (M), |m ii | > p (k−1) Proof. Let X be the diagonal matrix diag{x 1 , x 2 , · · · , x n }, where If j∈N 2 (M)\{i} |m i j | = 0, then the corresponding fraction is defined to be ∞. Next we consider the following two cases. Case 1: For each i ∈ N 1 (M), it is not difficult to see that |(MX) ii | = |m ii |, and Based on Cases 1 and 2, we have that MX is an S DD matrix, and consequently, M is an H-matrix. The proof is completed. □ According to the definition of S DD k matrix and Lemma 1, we obtain some properties of S DD k matrices as follows: Then Proof. By Lemma 1 and the known conditions that for ∀i ∈ N 2 (M), We now prove that |m ii | > p (k−1) i (M) > 0(∀i ∈ N) and consider the following two cases.
Case 2: If N 1 (M) ∅, by Theorem 5 and the known condition that for ∀i ∈ N 2 (M), Then there exists a diagonal matrix X = diag{x 1 , x 2 , · · · , x n } such that MX is an S DD matrix. Elements x 1 , x 2 , . . . , x n are determined by Proof. We need to prove that matrix MX satisfies the following inequalities: From Theorem 6 and the known condition that for ∀i ∈ N 2 (M), Therefore, MX is an S DD matrix. The proof is completed. □

Infinity norm upper bounds for the inverse of S DD and S DD k matrices
In this section, by Lemma 2 and Theorem 7, we provide some new upper bounds of the infinity norm of the inverse matrices for S DD k matrices and S DD matrices, respectively. We also present some comparisons with the Varah bound. Some numerical examples are presented to illustrate the corresponding results. Specially, when the involved matrices are S DD 1 matrices as subclass of S DD k matrices, these new bounds are in line with that provided by Chen et al. [20].
Proof. By Lemma 2, we have that there exists a positive diagonal matrix X such that MX is an S DD matrix, where X is defined as Lemma 2. Thus, Notice that MX is an S DD matrix. Hence, by Theorem 1, we have .
Thus, for any i ∈ N 1 (M), it holds that For any i ∈ N 2 (M), it holds that Therefore, we get The proof is completed. □ From Theorem 8, it is easy to obtain the following result.
Corollary 1. Let M = (m i j ) ∈ C n×n (n ≥ 2) be an S DD matrix. Then .
Example 1. Consider the n × n matrix: Take that n = 20. It is easy to verify that M is an S DD matrix. By calculations, we have that for k = 2, Moreover, when k = 1, by Theorem 2, we have ||M −1 || ∞ ≤ 0.7188 + ε 5 0.4844 + 0.5ε 5 , 0 < ε 5 < 0.3214. The following Theorem 9 shows that the bound in Corollary 1 is better than that in Theorem 1 of [20] in some cases.
where Z i and ε are defined as in Corollary 1, respectively.
Proof. From the given condition, we have that there exists k ≥ 2 such that Thus, we get Since M is an S DD matrix, then It's easy to verify that .
Thus, by Corollary 1, it holds that The proof is completed. □ We illustrate Theorem 9 by the following Example 2.
Example 2. This is the previous Example 1. For k = 4, we have Thus, by Theorem 8, we obtain that for each 0 < ε 2 < 0.7034, .
However, we find that the upper bounds in Theorems 8 and 9 contain the parameter ε. Next, based on Theorem 7, we will provide new upper bounds for the infinity norm of the inverse matrices of S DD k matrices, which only depend on the elements of the given matrices.
Theorem 10. Let M = (m i j ) ∈ C n×n (n ≥ 2) be an S DD k (k ≥ 2) matrix and for each i ∈ N 2 (M), Proof. By Theorems 7 and 8, we have that there exists a positive diagonal matrix X such that MX is an S DD matrix, where X is defined as in Theorem 7. Thus, it holds that Notice that MX is an S DD matrix. Thus, by Theorem 1, we get Therefore, we have that The proof is completed. □ Since S DD matrices are a subclass of S DD k matrices, by Theorem 10, we can obtain the following result.
Corollary 2. Let M = (m i j ) ∈ C n×n (n ≥ 2) be an S DD matrix. If r i (M) > 0(∀i ∈ N), then there exists k ≥ 2 such that Two examples are given to show the advantage of the bound in Theorem 10.
Example 3. Consider the following matrix: It is easy to verify that M is not an S DD matrix, an S DD 1 matrix, a GS DD 1 matrix, an S -S DD matrix, nor a CKV-type matrix. Therefore, we cannot use the error bounds in [1,8,9,18,20] to estimate ||M −1 || ∞ . But, M is an S DD 2 matrix. So by the bound in Theorem 10, we have that ∥M −1 ∥ ∞ ≤ 0.5820.
Example 4. Consider the tri-diagonal matrix M ∈ R n×n arising from the finite difference method for free boundary problems [18], where Take that n = 4, a = 1, b = 0 , c = 3.7 and α = 10. It is easy to verify that M is neither an S DD matrix nor an S DD 1 matrix. However, we can get that M is a GS DD 1 matrix and an S DD 3 matrix. By the bound in Theorem 10, we have ∥M −1 ∥ ∞ ≤ 8.2630, while by the bound in Theorem 4, it holds that , ε ∈ (1.4661, 2.1488).
The following two theorems show that the bound in Corollary 2 is better than that in Theorem 1 in some cases.
Theorem 11. Let M = (m i j ) ∈ C n×n (n ≥ 2) be an S DD matrix. If r i (M) > 0(∀i ∈ N) and there exists k ≥ 2 such that min i∈N j∈N\{i} .
Proof. Since M is an S DD matrix, then N 1 (M) = ∅ and M is an S DD k matrix. By the given condition that r i (M) > 0(∀i ∈ N), it holds that Similarly, we can obtain that Since there exists k ≥ 2 such that min i∈N j∈N\{i} .
Thus, from Corollary 2, we get .
The proof is completed. □ We illustrate the Theorem 11 by following Example 5.
Take that n = 20. It is easy to check that M is an S DD matrix. Let Thus, when k = 2, 3, 4, 5, 6, the matrix M satisfies the conditions of Theorem 11. By Theorems 1 and 11, we can derive the upper bounds for ||M −1 || ∞ , see Table 2. Meanwhile, when k = 1, by Theorem 3, we get that ||M −1 || ∞ ≤ 1.6976. From Table 2, we can see that the bounds in Theorem 11 are better than that in Theorems 1 and 3 in some cases.
Theorem 12. Let M = (m i j ) ∈ C n×n (n ≥ 2) be an S DD matrix. If r i (M) > 0(∀i ∈ N) and there exists k ≥ 2 such that .
Proof. By Theorem 7 and the given condition that r i (M) > 0(∀i ∈ N), it is easy to get that From the condition that there exists k ≥ 2 such that Thus, from Corollary 2, it holds that .
The proof is completed. □ Next, we illustrate Theorem 12 by the following Example 6.
Take that n = 20. It is easy to verify that M is an S DD matrix.
From Table 3, we can see that the bound in Theorem 12 is sharper than that in Theorems 1 and 3 in some cases.

Conclusions
S DD k matrices as a new subclass of H-matrices are proposed, which include S DD matrices and S DD 1 matrices, and some properties of S DD k matrices are obtained. Meanwhile, some new upper bounds of the infinity norm of the inverse matrices for S DD matrices and S DD k matrices are presented. Furthermore, we prove that the new bounds are better than some existing bounds in some cases. Some numerical examples are also provided to show the validity of new results. In the future, based on the proposed infinity norm bound, we will explore the computable error bounds of the linear complementarity problems for S DD k matrices.

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