2 | 4 | 6 | 8 | |
Cor |
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Th |
2 | 2 | 2 | 2 |
In this paper, we introduce a new subclass of H-matrices called SDDk matrices, which contains SDD matrices and SDD1 matrices as special cases, and present some properties of SDDk matrices. Based on these properties, we also provide new infinity norm bounds for the inverse of SDD matrices and SDDk matrices. It is proved that these new bounds are better than some existing results in some cases. Numerical examples demonstrate the effectiveness of the obtained results.
Citation: Xiaodong Wang, Feng Wang. Infinity norm upper bounds for the inverse of SDDk matrices[J]. AIMS Mathematics, 2023, 8(10): 24999-25016. doi: 10.3934/math.20231276
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In this paper, we introduce a new subclass of H-matrices called SDDk matrices, which contains SDD matrices and SDD1 matrices as special cases, and present some properties of SDDk matrices. Based on these properties, we also provide new infinity norm bounds for the inverse of SDD matrices and SDDk matrices. It is proved that these new bounds are better than some existing results in some cases. Numerical examples demonstrate the effectiveness of the obtained results.
Let n be an integer number, N={1,2,…,n}, and let Cn×n be the set of all complex matrices of order n. A matrix M=(mij)∈Cn×n(n≥2) is called a strictly diagonally dominant (SDD) matrix [1] if
|mii|>ri(M)=n∑j=1,j≠i|mij|, ∀i∈N. |
It was shown that an SDD matrix is an H-matrix [1], where matrix M=(mij)∈Cn×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 SDD 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-SDDS matrices [9], DZ and DZ-type matrices [10, 11], Nekrasov matrices [12, 13, 14, 15], S-Nekrasov matrices [16], Q-Nekrasov matrices [17], GSDD1 matrices [18] and so on.
In 2011, Peňa [19] proposed a new subclass of H-matrices called SDD1 matrices, whose definition is listed below. A matrix M=(mij)∈Cn×n(n≥2) is called an SDD1 matrix if
|mii|>pi(M),∀i∈N1(M), |
where
pi(M)=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}rj(M)|mjj||mij|,N1(M)={i||mii|≤ri(M)}, N2(M)={i||mii|>ri(M)}. |
In 2023, Dai et al. [18] gave a new subclass of H-matrices named generalized SDD1 (GSDD1) matrices, which extends the class of SDD1 matrices. Here, a matrix M=(mij)∈Cn×n(n≥2) is said a GSDD1 matrix if
ri(M)−pN2(M)i(M)>0,∀i∈N2(M), |
and
(ri(M)−pN2(M)i(M))(|ajj|−pN1(M)j(M))>pN1(M)i(M)pN2(M)j(M),∀i∈N2(M),∀j∈N1(M), |
where
pN2(M)i(M)=∑j∈N2(M)∖{i}rj(M)|mjj||mij|,pN1(M)i(M)=∑j∈N1(M)∖{i}|mij|,i∈N. |
Subsequently, some upper bounds for the infinite norm of the inverse matrices of SDD matrices, SDD1 matrices and GSDD1 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=(mij)∈Cn×n(n≥2) be an SDD matrix. Then
||M−1||∞≤1min1≤i≤n(|mii|−ri(M)). |
Theorem 2. [20] Let matrix M=(mij)∈Cn×n(n≥2) be an SDD matrix. Then
||M−1||∞≤maxi∈Npi(M)|mii|+εmini∈NZi,0<ε<mini∈N|mii|−pi(M)ri(M), |
where
Zi=ε(|mii|−ri(M))+∑j∈N∖{i}(rj(M)−pj(M)|mjj|)|mij|. |
Theorem 3. [20] Let matrix M=(mij)∈Cn×n(n≥2) be an SDD matrix. If ri(M)>0(∀i∈N), then
||M−1||∞≤maxi∈Npi(M)|mii|mini∈N∑j∈N∖{i}rj(M)−pj(M)|mjj||mij|. |
Theorem 4. [18] Let M=(mij)∈Cn×n be a GSDD1 matrix. Then
||M−1||∞≤max{ε,maxi∈N2(M)ri(M)|mii|}min{mini∈N2(M)ϕi,mini∈N1(M)ψi}, |
where
ϕi=ri(M)−∑j∈N2(M)∖{i}rj(M)|mjj||mij|−∑j∈N1(M)∖{i}|mij|ε,ψi=|mii|ε−∑j∈N1(M)∖{i}|mij|ε+∑j∈N2(M)∖{i}rj(M)|mjj||mij|, |
and
maxi∈N1(M)pN2(M)i(M)|mii|−pN1(M)i(M)<ε<minj∈N2(M)rj(M)−pN2(M)j(M)pN1(M)j(M). |
On the basis of the above articles, we continue to study special structured matrices and introduce a new subclass of H-matrices called SDDk matrices, and provide some new upper bounds for the infinite norm of the inverse matrices for SDD matrices and SDDk 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 SDDk matrices, which include SDD matrices and SDD1 matrices, and derive some properties of SDDk matrices. In Section 3, we present some upper bounds for the infinity norm of the inverse matrices for SDD matrices and SDDk matrices, and provide some comparisons with the well-known Varah bound. Moreover, some numerical examples are given to illustrate the corresponding results.
In this section, we propose a new subclass of H-matrices called SDDk matrices, which include SDD matrices and SDD1 matrices, and derive some new properties.
Definition 1. A matrix M=(mij)∈Cn×n(n≥2) is called an SDDk matrix if there exists k∈N such that
|mii|>p(k−1)i(M),∀i∈N1(M), |
where
p(k)i(M)=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(k−1)j(M)|mjj||mij|,p(0)i(M)=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}rj(M)|mjj||mij|. |
Immediately, we know that SDDk matrices contain SDD matrices and SDD1 matrices, so
{SDD}⊆{SDD1}⊆{SDD2}⊆⋯⊆{SDDk}. |
Lemma 1. A matrix M=(mij)∈Cn×n(n≥2) is an SDDk(k≥2) matrix if and only if for ∀i∈N, |mii|>p(k−1)i(M).
Proof. For ∀i∈N1(M), from Definition 1, it holds that |mii|>p(k−1)i(M).
For ∀i∈N2(M), we have that |mii|>ri(M). When k=2, it follows that
|mii|>ri(M)≥∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}rj(M)|mjj||mij|=p(0)i(M)≥∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(0)j(M)|mjj||mij|=p(1)i(M). |
Suppose that |mii|>p(k−1)i(M)(k≤l,l>2) holds for ∀i∈N2(M). When k=l+1, we have
|mii|>ri(M)≥∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(l−2)j(M)|mjj||mij|=p(l−1)i(M)≥∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(l−1)j(M)|mjj||mij|=p(l)i(M). |
By induction, we obtain that for ∀i∈N2(M), |mii|>p(k−1)i(M). Consequently, it holds that matrix M is an SDDk matrix if and only if |mii|>p(k−1)i(M) for ∀i∈N. The proof is completed.
Lemma 2. If M=(mij)∈Cn×n(n≥2) is an SDDk(k≥2) matrix, then M is an H-matrix.
Proof. Let X be the diagonal matrix diag{x1,x2,⋯,xn}, where
(0<) xj={1,j∈N1(M),p(k−1)j(M)|mjj|+ε,j∈N2(M), |
and
0<ε<mini∈N|mii|−p(k−1)i(M)∑j∈N2(M)∖{i}|mij|. |
If ∑j∈N2(M)∖{i}|mij|=0, then the corresponding fraction is defined to be ∞. Next we consider the following two cases.
Case 1: For each i∈N1(M), it is not difficult to see that |(MX)ii|=|mii|, and
ri(MX)=∑j=1,j≠i|mij|xj=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}(p(k−1)j(M)|mjj|+ε)|mij|≤∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}(p(k−2)j(M)|mjj|+ε)|mij|=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(k−2)j(M)|mjj||mij|+∑j∈N2(M)∖{i}ε|mij|=p(k−1)i(M)+ε∑j∈N2(M)∖{i}|mij|<p(k−1)i(M)+|mii|−p(k−1)i(M)=|mii|=|(MX)ii|. |
Case 2: For each i∈N2(M), we can obtain that
|(MX)ii|=|mii|(pk−1i(M)|mii|+ε)=p(k−1)i(M)+ε|mii|, |
and
ri(MX)=∑j=1,j≠i|mij|xj=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}(p(k−1)j(M)|mjj|+ε)|mij|≤∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}(p(k−2)j(M)|mjj|+ε)|mij|=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(k−2)j(M)|mjj||mij|+∑j∈N2(M)∖{i}ε|mij|=p(k−1)i(M)+ε∑j∈N2(A)∖{i}|mij|<p(k−1)i(M)+ε|mii|=|(MX)ii|. |
Based on Cases 1 and 2, we have that MX is an SDD matrix, and consequently, M is an H-matrix. The proof is completed.
According to the definition of SDDk matrix and Lemma 1, we obtain some properties of SDDk matrices as follows:
Theorem 5. If M=(mij)∈Cn×n(n≥2) is an SDDk matrix and N1(M)≠∅, then for ∀i∈N1(M), ∑j≠i,j∈N2(M)|mij|>0.
Proof. Suppose that for ∀i∈N1(M), ∑j≠i,j∈N2(M)|mij|=0. By Definition 1, we have that p(k−1)i(M)=ri(M), ∀i∈N1(M). Thus, it is easy to verify that |mii|>p(k−1)i(M)=ri(M)≥|mii|, which is a contradiction. Thus for ∀i∈N1(M), ∑j≠i,j∈N2(M)|mij|>0. The proof is completed.
Theorem 6. Let M=(mij)∈Cn×n(n≥2) be an SDDk(k≥2) matrix. It holds that ∑j≠i,j∈N2(M)|mij|>0, ∀i∈N2(M). Then
|mii|>p(k−2)i(M)>p(k−1)i(M)>0,∀i∈N2(M), |
and
|mii|>p(k−1)i(M)>0,∀i∈N. |
Proof. By Lemma 1 and the known conditions that for ∀i∈N2(M), ∑j≠i,j∈N2(M)|mij|>0, it holds that
|mii|>p(k−2)i(M)>p(k−1)i(M)>0,∀i∈N2(M), |
and
|mii|>p(k−1)i(M),∀i∈N. |
We now prove that |mii|>p(k−1)i(M)>0(∀i∈N) and consider the following two cases.
Case 1: If N1(M)=∅, then M is an SDD matrix. It is easy to verify that |mii|>p(k−1)i(M)>0, ∀i∈N2(M)=N.
Case 2: If N1(M)≠∅, by Theorem 5 and the known condition that for ∀i∈N2(M), ∑j≠i,j∈N2(M)|mij|>0, then it is easy to obtain that |mii|>p(k−1)i(M)>0(∀i∈N).
From Cases 1 and 2, we have that |mii|>p(k−1)i(M)>0(∀i∈N). The proof is completed.
Theorem 7. Let M=(mij)∈Cn×n(n≥2) be an SDDk(k≥2) matrix and for ∀i∈N2(M), ∑j≠i,j∈N2(M)|mij|>0. Then there exists a diagonal matrix X=diag{x1,x2,⋯,xn} such that MX is an SDD matrix. Elements x1,x2,…,xn are determined by
xi=p(k−1)i(M)|mii|,∀i∈N. |
Proof. We need to prove that matrix MX satisfies the following inequalities:
|(MX)ii|>ri(MX),∀i∈N. |
From Theorem 6 and the known condition that for ∀i∈N2(M), ∑j≠i,j∈N2(M)|mij|>0, it is easy to verify that
0<p(k−1)i(M)|mii|<p(k−2)i(M)|mii|<1,∀i∈N2(M). |
For each i∈N, we have that |(MX)ii|=p(k−1)i(M) and
ri(MX)=∑j=1,j≠i|mij|xj=∑j∈N1(M)∖{i}p(k−1)j(M)|mjj||mij|+∑j∈N2(M)∖{i}p(k−1)j(M)|mjj||mij|<∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(k−2)j(M)|mjj||mij|=p(k−1)i(M)=|(MX)ii|, |
that is,
|(MX)ii|>ri(MX). |
Therefore, MX is an SDD matrix. The proof is completed.
In this section, by Lemma 2 and Theorem 7, we provide some new upper bounds of the infinity norm of the inverse matrices for SDDk matrices and SDD 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 SDD1 matrices as subclass of SDDk matrices, these new bounds are in line with that provided by Chen et al. [20].
Theorem 8. Let M=(mij)∈Cn×n(n≥2) be an SDDk(k≥2) matrix. Then
||M−1||∞≤max{1,maxi∈N2(M)p(k−1)i(M)|mii|+ε}min{mini∈N1(M)Ui,mini∈N2(M)Vi}, |
where
Ui=|mii|−∑j∈N1(M)∖{i}|mij|−∑j∈N2(M)∖{i}(p(k−1)j(M)|mjj|+ε)|mij|,Vi=ε(|mii|−∑j∈N2(M)∖{i}|mij|)+∑j∈N2(M)∖{i}(p(k−2)j(M)−p(k−1)j(M)|mjj|)|mij|, |
and
0<ε<mini∈N|mii|−p(k−1)i(M)∑j∈N2(M)∖{i}|mij|. |
Proof. By Lemma 2, we have that there exists a positive diagonal matrix X such that MX is an SDD matrix, where X is defined as Lemma 2. Thus,
||M−1||∞=||X(X−1M−1)||∞=||X(MX)−1||∞≤||X||∞||(MX)−1||∞, |
and
||X||∞=max1≤i≤nxi=max{1,maxi∈N2(M)p(k−1)i(M)|mii|+ε}. |
Notice that MX is an SDD matrix. Hence, by Theorem 1, we have
||(MX)−1||∞≤1min1≤i≤n(|(MX)ii|−ri(MX)). |
Thus, for any i∈N1(M), it holds that
|(MX)ii|−ri(MX)=|mii|−∑j∈N1(M)∖{i}|mij|−∑j∈N2(M)∖{i}(p(k−1)j(M)|mjj|+ε)|mij|=Ui. |
For any i∈N2(M), it holds that
|(MX)ii|−ri(MX)=p(k−1)i(M)+ε|mii|−∑j∈N1(M)∖{i}|mij|−∑j∈N2(M)∖{i}(p(k−1)j(M)|mjj|+ε)|mij|=∑j∈N1(M)∖{i}|mij|+∑j∈N2(M)∖{i}p(k−2)j(M)|mjj||mij|+ε|mii|−∑j∈N1(M)∖{i}|mij|−∑j∈N2(M)∖{i}(p(k−1)j(M)|mjj|+ε)|mij|=ε(|mii|−∑j∈N2(M)∖{i}|mij|)+∑j∈N2(M)∖{i}(p(k−2)j(M)−p(k−1)j(M)|mjj|)|mij|=Vi. |
Therefore, we get
||M−1||∞≤max{1,maxi∈N2(M)p(k−1)i(M)|mii|+ε}min{mini∈N1(M)Xi,mini∈N2(M)Yi}. |
The proof is completed.
From Theorem 8, it is easy to obtain the following result.
Corollary 1. Let M=(mij)∈Cn×n(n≥2) be an SDD matrix. Then
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|+εmini∈NZi, |
where k≥2,
Zi=ε(|mii|−ri(M))+∑j∈N∖{i}(p(k−2)j(M)−p(k−1)j(M)|mjj|)|mij|, |
and
0<ε<mini∈N|mii|−p(k−1)i(M)ri(M). |
Example 1. Consider the n×n matrix:
M=(421.51.542482482⋱⋱⋱4824824823.54). |
Take that n=20. It is easy to verify that M is an SDD matrix.
By calculations, we have that for k=2,
maxi∈Np(1)i(M)|mii|+ε1=0.5859+ε1,mini∈NZi=0.4414+0.5ε1,0<ε1<0.4732. |
For k=4,
maxi∈Np(3)i(M)|aii|+ε2=0.3845+ε2,mini∈NZi=0.2959+0.5ε2,0<ε2<0.7034. |
For k=6,
maxi∈Np(5)i(M)|mii|+ε3=0.2504+ε3,mini∈NZi=0.1733+0.5ε3,0<ε3<0.8567. |
For k=8,
maxi∈Np(7)i(M)|mii|+ε4=0.1624+ε4,mini∈NZi=0.0990+0.5ε4,0<ε4<0.9572. |
So, when k=2,4,6,8, by Corollary 1 and Theorem 1, we can get the upper bounds for ||M−1||∞, see Table 1. Thus,
||M−1||∞≤0.5859+ε10.4414+0.5ε1<2,||M−1||∞≤0.3845+ε20.2959+0.5ε2<2, |
2 | 4 | 6 | 8 | |
Cor |
||||
Th |
2 | 2 | 2 | 2 |
and
||M−1||∞≤0.2504+ε30.1733+0.5ε3<2,||M−1||∞≤0.1624+ε40.0990+0.5ε4<2. |
Moreover, when k=1, by Theorem 2, we have
||M−1||∞≤0.7188+ε50.4844+0.5ε5,0<ε5<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.
Theorem 9. Let matrix M=(mij)∈Cn×n(n≥2) be an SDD matrix. If there exists k≥2 such that
maxi∈Np(k−1)i(M)|mii|mini∈N(|mii|−ri(M))≤mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|, |
then
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|+εmini∈NZi≤1min1≤i≤n(|mii|−ri(M)), |
where Zi and ε are defined as in Corollary 1, respectively.
Proof. From the given condition, we have that there exists k≥2 such that
maxi∈Np(k−1)i(M)|mii|mini∈N(|mii|−ri(M))≤mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|, |
then
maxi∈Np(k−1)i(M)|mii|mini∈N(|mii|−ri(M))+εmini∈N(|mii|−ri(M))≤mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|+εmini∈N(|mii|−ri(M)). |
Thus, we get
(maxi∈Np(k−1)i(M)|mii|+ε)mini∈N(|mii|−ri(M))≤mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|+εmini∈N(|mii|−ri(M))=mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|+mini∈N(ε(|mii|−ri(M)))≤mini∈N(ε(|mii|−ri(M))+∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|)=mini∈NZi. |
Since M is an SDD matrix, then
|mii|>ri(M),Zi>0,∀i∈N. |
It's easy to verify that
maxi∈Np(k−1)i(M)|mii|+εmini∈NZi≤1min1≤i≤n(|mii|−ri(M)). |
Thus, by Corollary 1, it holds that
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|+εmini∈NZi≤1min1≤i≤n(|mii|−ri(M)). |
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
maxi∈Np(3)i(M)|mii|mini∈N(|mii|−ri(M))=0.1923<0.2959=mini∈N∑j∈N∖{i}p(2)j(M)−p(3)j(M)|mjj||mij|. |
Thus, by Theorem 8, we obtain that for each 0<ε2<0.7034,
||M−1||∞≤0.3845+ε20.2959+0.5ε2<2=1min1≤i≤n(|mii|−ri(M)). |
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 SDDk matrices, which only depend on the elements of the given matrices.
Theorem 10. Let M=(mij)∈Cn×n(n≥2) be an SDDk(k≥2) matrix and for each i∈N2(M), ∑j≠i,j∈N2(M)|mij|>0. Then
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N(p(k−1)i(M)−∑j∈N∖{i}p(k−1)j(M)|mjj||mij|). |
Proof. By Theorems 7 and 8, we have that there exists a positive diagonal matrix X such that MX is an SDD matrix, where X is defined as in Theorem 7. Thus, it holds that
||M−1||∞=||X(X−1M−1)||∞=||X(MX)−1||∞≤||X||∞||(MX)−1||∞, |
and
||X||∞=max1≤i≤nxi=maxi∈Np(k−1)i(M)|mii|. |
Notice that MX is an SDD matrix. Thus, by Theorem 1, we get
||(MX)−1||∞≤1min1≤i≤n(|(MX)ii|−ri(MX))=1min1≤i≤n(|miixi|−ri(MX))=1mini∈N(p(k−1)i(M)−∑j∈N∖{i}p(k−1)j(M)|mjj||mij|). |
Therefore, we have that
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N(p(k−1)i(M)−∑j∈N∖{i}p(k−1)j(M)|mjj||mij|). |
The proof is completed.
Since SDD matrices are a subclass of SDDk matrices, by Theorem 10, we can obtain the following result.
Corollary 2. Let M=(mij)∈Cn×n(n≥2) be an SDD matrix. If ri(M)>0(∀i∈N), then there exists k≥2 such that
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|. |
Two examples are given to show the advantage of the bound in Theorem 10.
Example 3. Consider the following matrix:
M=(40−1−2−1−2010−4.1−4−6−20−233−4−80−4−620−2−30−4−2040). |
It is easy to verify that M is not an SDD matrix, an SDD1 matrix, a GSDD1 matrix, an S-SDD 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 SDD2 matrix. So by the bound in Theorem 10, we have that ‖M−1‖∞≤0.5820.
Example 4. Consider the tri-diagonal matrix M∈Rn×n arising from the finite difference method for free boundary problems [18], where
M=(b+αsin(1n)c0⋯0ab+αsin(2n)c⋯0⋱⋱⋱0⋯ab+αsin(n−1n)c0⋯0ab+αsin(1)). |
Take that n=4, a=1, b=0, c=3.7 and α=10. It is easy to verify that M is neither an SDD matrix nor an SDD1 matrix. However, we can get that M is a GSDD1 matrix and an SDD3 matrix. By the bound in Theorem 10, we have
‖M−1‖∞≤8.2630, |
while by the bound in Theorem 4, it holds that
‖M−1‖∞≤εmin{2.1488−ε,0.3105,2.474ε−3.6272},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=(mij)∈Cn×n(n≥2) be an SDD matrix. If ri(M)>0(∀i∈N) and there exists k≥2 such that
mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|≥mini∈N(|mii|−ri(M)), |
then
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|<1min1≤i≤n(|mii|−ri(M)). |
Proof. Since M is an SDD matrix, then N1(M)=∅ and M is an SDDk matrix. By the given condition that ri(M)>0(∀i∈N), it holds that
|mii|>ri(M)>∑j∈N∖{i}rj(M)|mjj||mij|=p(0)i(M)>0,∀i∈N,p(0)i(M)=∑j∈N∖{i}rj(M)|mjj||mij|>∑j∈N∖{i}p(0)j(M)|mjj||mij|=p(1)i(M)>0,∀i∈N. |
Similarly, we can obtain that
|mii|>ri(M)>p(0)i(M)>⋯>p(k−1)i(M)>0,∀i∈N, |
that is,
maxi∈Np(k−1)i(M)|mii|<1. |
Since there exists k≥2 such that
mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|≥mini∈N(|mii|−ri(M)), |
then we have
maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|<1min1≤i≤n(|mii|−ri(M)). |
Thus, from Corollary 2, we get
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|<1min1≤i≤n(|mii|−ri(M)). |
The proof is completed.
We illustrate the Theorem 11 by following Example 5.
Example 5. Consider the matrix M=(mij)∈Cn×n(n≥2), where
M=(430.9162252252⋱⋱⋱2522521620.934). |
Take that n=20. It is easy to check that M is an SDD matrix. Let
lk=mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|,m=mini∈N(|mii|−ri(M)). |
By calculations, we have
l2=0.2692>0.1=m,l3=0.2567>0.1=m,l4=0.1788>0.1=m,l5=0.1513>0.1=m,l6=0.1037>0.1=m. |
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.
2 | 3 | 4 | 5 | 6 | |
Th 11 | 1.9022 | 1.5959 | 1.8332 | 1.7324 | 2.0214 |
Th 1 | 10 | 10 | 10 | 10 | 10 |
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=(mij)∈Cn×n(n≥2) be an SDD matrix. If ri(M)>0(∀i∈N) and there exists k≥2 such that
maxi∈Np(k−1)i(M)|mii|mini∈N(|mii|−ri(M))≤mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|<mini∈N(|mii|−ri(M)), |
then
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|≤1min1≤i≤n(|mii|−ri(M)). |
Proof. By Theorem 7 and the given condition that ri(M)>0(∀i∈N), it is easy to get that
∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|>0,∀i∈N. |
From the condition that there exists k≥2 such that
maxi∈Np(k−1)i(M)|mii|mini∈N(|mii|−ri(M))≤mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|, |
we have
maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|≤1min1≤i≤n(|mii|−ri(M)). |
Thus, from Corollary 2, it holds that
||M−1||∞≤maxi∈Np(k−1)i(M)|mii|mini∈N∑j∈N∖{i}p(k−2)j(M)−p(k−1)j(M)|mjj||mij|≤1min1≤i≤n(|mii|−ri(M)). |
The proof is completed.
Next, we illustrate Theorem 12 by the following Example 6.
Example 6. Consider the tri-diagonal matrix M=(mij)∈Cn×n(n≥2), where
M=(3−2.5−1.24−2−2.85−1−2.85−1⋱⋱⋱−2.85−1−2.85−1−1.24−2−2.53). |
Take that n=20. It is easy to verify that M is an SDD matrix.
By calculations, we have that for k=2,
maxi∈Np(1)i(M)|mii|mini∈N(|mii|−ri(M))=0.2686<mini∈N∑j∈N∖{i}p(0)j(M)−p(1)j(M)|mjj||mij|=0.3250<0.5=mini∈N(|mii|−ri(M)). |
For k=5, we get
maxi∈Np(4)i(M)|mii|mini∈N(|mii|−ri(M))=0.1319<mini∈N∑j∈N∖{i}p(3)j(M)−p(4)j(M)|mjj||mij|=0.1685<0.5=mini∈N(|mii|−ri(M)). |
For k=10, it holds that
maxi∈Np(9)i(M)|mii|mini∈N(|mii|−ri(M))=0.0386<mini∈N∑j∈N∖{i}p(8)j(M)−p(9)j(M)|mjj||mij|=0.0485<0.5=mini∈N(|mii|−ri(M)). |
Thus, for k=2,5,10, the matrix M satisfies the conditions of Theorem 12. Thus, from Theorems 12 and 1, we get the upper bounds for ||M−1||∞, see Table 3. Meanwhile, when k=1, by Theorem 3, we have that ||M−1||∞≤1.7170.
2 | 5 | 10 | |
Th |
1.6530 | 1.5656 | 1.5925 |
Th |
2 | 2 | 2 |
From Table 3, we can see that the bound in Theorem 12 is sharper than that in Theorems 1 and 3 in some cases.
SDDk matrices as a new subclass of H-matrices are proposed, which include SDD matrices and SDD1 matrices, and some properties of SDDk matrices are obtained. Meanwhile, some new upper bounds of the infinity norm of the inverse matrices for SDD matrices and SDDk 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 SDDk matrices.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research is supported by Guizhou Provincial Science and Technology Projects (20191161), and the Natural Science Research Project of Department of Education of Guizhou Province (QJJ2023062, QJJ2023063).
The authors declare that they have no competing interests.
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2 | 4 | 6 | 8 | |
Cor |
||||
Th |
2 | 2 | 2 | 2 |
2 | 3 | 4 | 5 | 6 | |
Th 11 | 1.9022 | 1.5959 | 1.8332 | 1.7324 | 2.0214 |
Th 1 | 10 | 10 | 10 | 10 | 10 |
2 | 5 | 10 | |
Th |
1.6530 | 1.5656 | 1.5925 |
Th |
2 | 2 | 2 |