Research article

CKV-type B-matrices and error bounds for linear complementarity problems

  • Received: 11 May 2021 Accepted: 16 July 2021 Published: 27 July 2021
  • MSC : 15A24, 15A60, 90C33, 65G50

  • In this paper, we introduce a new subclass of P-matrices called Cvetković-Kostić-Varga type B-matrices (CKV-type B-matrices), which contains DZ-type-B-matrices as a special case, and present an infinity norm bound for the inverse of CKV-type B-matrices. Based on this bound, we also give an error bound for linear complementarity problems of CKV-type B-matrices. It is proved that the new error bound is better than that provided by Li et al. [24] for DZ-type-B-matrices, and than that provided by M. García-Esnaola and J.M. Peña [10] for B-matrices in some cases. Numerical examples demonstrate the effectiveness of the obtained results.

    Citation: Xinnian Song, Lei Gao. CKV-type B-matrices and error bounds for linear complementarity problems[J]. AIMS Mathematics, 2021, 6(10): 10846-10860. doi: 10.3934/math.2021630

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  • In this paper, we introduce a new subclass of P-matrices called Cvetković-Kostić-Varga type B-matrices (CKV-type B-matrices), which contains DZ-type-B-matrices as a special case, and present an infinity norm bound for the inverse of CKV-type B-matrices. Based on this bound, we also give an error bound for linear complementarity problems of CKV-type B-matrices. It is proved that the new error bound is better than that provided by Li et al. [24] for DZ-type-B-matrices, and than that provided by M. García-Esnaola and J.M. Peña [10] for B-matrices in some cases. Numerical examples demonstrate the effectiveness of the obtained results.



    Given an n×n real matrix A and a vector qRn, the linear complementarity problem is to find a vector xRn satisfying

    x0,Ax+q0,(Ax+q)Tx=0 (1.1)

    or to show that no such vector x exists. We denote the problem (1.1) and its solution by LCP(A,q) and x, respectively. The LCP(A,q), as one of the fundamental problems in optimization and mathematical programming, has various applications in the quadratic programming, the optimal stopping, the Nash equilibrium point of a bimatrix game, the network equilibrium problem, the contact problem, and the free boundary problem for journal bearing, for details, see [1,2,26].

    It is well known that the LCP(A,q) has a unique solution x for any qRn if and only if A is a P-matrix [2]. Here, a real square matrix A is called a P-matrix if all its principal minors are positive. For this case, an important topic in the study of the LCP(A,q) concerns the bound of ||xx||, since it can be used as termination criteria for iterative algorithms and can be used to measure the sensitivity of the solution of LCP(A,q) in response to a small perturbation, e.g., [18,19,28,34]. When the matrix A is a P-matrix, Chen and Xiang [3] gave the following error bound for the LCP(A,q):

    ||xx||maxd[0,1]n||(ID+DA)1||||r(x)||, (1.2)

    where D=diag(di) with 0di1 for each iN:={1,,n}, d=(d1,d2,,dn)T[0,1]n, and r(x)=min{x,Ax+q} in which the min operator denotes the componentwise minimum of two vectors. Furthermore, to avoid the high-cost computations of the inverse matrix in (1.2), some easily computable bounds for the LCP(A,q) were derived for the different subclass of P-matrices, such as, B-matrices [10,20], doubly B-matrices [6], SB-matrices [7,8], MB-matrices [4], B-Nekrasov matrices [11,21], weakly chained diagonally dominant B-matrices [22,32,35], BRπ-matrices [12,13,27], and so on [9,14,15,16,17,23,36].

    Recently, Li et al. [24] presented a new subclass of P-matrices called Dashnic-Zusmanovich type B-matrices (DZ-type-B-matrices), and provided an error bound for the LCP(A,q) when A is a DZ-type-B-matrix.

    Definition 1.1. [33] A matrix A=[aij]Cn×n, with n2, is a DZ-type matrix if for each iN, there exists jN,ji such that

    (|aii|rji(A))|ajj|>|aij|rj(A),

    where rji(A)=ri(A)|aij| and ri(A)=jN{i}|aij|.

    Definition 1.2. [24] Let A=[aij]Cn×n be written in the form

    A=B++C, (1.3)

    where

    B+=[bij]=[a11r+1a1nr+1an1r+nannr+n],C=[r+1r+1r+nr+n],

    and r+i:=max{0,aij|ji}. Then, A is called a DZ-type-B-matrix if B+ is a DZ-type matrix with all positive diagonal entries.

    Theorem 1.1. [24,Theorem 6] Let A=[aij]Rn×n be a DZ-type-B-matrix, and B+=[bij] be the matrix of (1.3). Then,

    maxd[0,1]n||(ID+DA)1||(n1)maxiNminjγi(B+)ζij(B+), (1.4)

    where

    γi(B+):={jN{i}:(|bii|rji(B+))|bjj|>|bij|rj(B+)},

    and

    ζij(B+):=(biirji(B+))bjjmax{1biirji(B+),1}+bjj|bij|max{1bjj,1}(biirji(B+))bjj|bij|rj(B+).

    Very recently, Cvetković et al. [5] proposed a new subclass of H-matrices called CKV-type matrices, which generalizes CKV matrices (also known as Σ-SDD matrices in the literature) and DZ-type matrices.

    Definition 1.3. [5] A matrix A=[aij]Cn×n, with n2, is called a CKV-type matrix if NA= or Si(A) is not empty for all iNA, where NA:={iN:|aii|ri(A)} and

    Si(A):={SΣ(i):|aii|>rSi(A),andforallj¯S(|aii|rSi(A))(|ajj|r¯Sj(A))>r¯Si(A)rSj(A)}

    with Σ(i):={SN:iS} and rSi(A):=jS{i}|aij|.

    Motivated by the definition of DZ-type-B-matrices, two meaningful questions naturally arise: can we get a more general subclass of P-matrices using CKV-type matrices, and can we obtain a sharper error bound than the bound (1.4) for the linear complementarity problem of DZ-type-B-matrices? To answer these questions, in Section 2, we present a new class of matrices: CKV-type B-matrices, and prove that it is a subclass of P-matrices containing DZ-type-B-matrices and SB-matrices. Meanwhile, we give an upper bound for the infinity norm for the inverse of CKV-type B-matrices. In Section 3, we give an error bound for the LCP(A,q) when A is a CKV-type B-matrix, consequently, for the LCP(A,q) when A is a DZ-type-B-matrix, and some comparisons with other results are also discussed. Finally, in Section 4, numerical examples are given to illustrate the corresponding theoretical results.

    Using CKV-type matrices, we first give the definition of CKV-type B-matrices.

    Definition 2.1. A matrix ARn×n is called a CKV-type B-matrix if B+ given by (1.3) is a CKV-type matrix with positive diagonal entries.

    To show that a CKV-type B-matrix is a P-matrix, we recall the following results.

    Lemma 2.1. [5,Theorem 6] Every CKV-type matrix is a nonsingular H-matrix.

    Lemma 2.2. [29,Corollary 2.4] If A is a real nonsingular M-matrix and P is a nonnegative matrix with rank(P) = 1, then A+P is a P-matrix.

    Proposition 2.1. If A is a CKV-type B-matrix, then A is a P-matrix.

    Proof. Let A be written in the form A=B++C as shown in (1.3). It follows from (1.3) and Definition 2.1 that B+ is a Z-matrix (all non-diagonal entries are non-positive [1]) with positive diagonal entries and C is a nonnegative matrix of rank 1. By Lemma 2.1, we know that B+ is a nonsingular H-matrix, and thus the conclusion follows from Lemma 2.2.

    As shown in [5,33], the relations of strictly diagonally dominant (SDD) matrices, doubly strictly diagonally dominant (DSDD) matrices, S-strictly diagonally dominant (S-SDD) matrices, DZ-type matrices, and CKV-type matrices are:

    {SDD}{DSDD}{DZ}{SSDD} and {SDD}{DZ-type};

    {DSDD}{DZ-type} and {DZ-type} {DSDD};

    {DZ}{DZ-type} and {DZ-type} {DZ};

    {S-SDD}{CKV -type} and {DZ-type} { CKV-type}.

    According to [24] and the above relations, we give a figure to illustrate the relations among B-matrices, DZ-type-B-matrices, DB-matrices, SB-matrices, CKV-type B-matrices. Here, the notions of B-matrices, DB-matrices, and of SB-matrices are listed as follows. Let A=B++CRn×n, where B+ is defined by (1.3). Then, A is called

    ● a B-matrix if B+ is SDD with all positive diagonal entries [30];

    ● a DB-matrix if B+ is DSDD with all positive diagonal entries [29];

    ● a SB-matrix if B+ is S-SDD with all positive diagonal entries for a given non-empty proper subset S of N [25].

    Figure 1.  Relations of CKV-type B-matrices and some existing subclasses of P-matrices.

    Next, we give a sufficient and necessary condition for a CKV-type B-matrix. Before that, a lemma is needed.

    Lemma 2.3. [5,Remark 9] A matrix A=[aij]Cn×n, with n2, is called a CKV-type matrix if Si(A) given by Definition 1.3 is not empty for all iN. Especially, if A is an SDD matrix, then for all iN, all proper subsets S containing i belong to Si(A).

    Proposition 2.2. Given any diagonal matrix D=diag(d1,d2,,dn) with di[0,1] for all iN, and let I be the identity matrix, then A is a CKV-type B-matrix if and only if ID+DA is a CKV-type B-matrix.

    Proof. Sufficiency is clearly established. We next show the necessity. Suppose A=[aij]Rn×n is a CKV-type B-matrix. Then, A=B++C, where B+ is the matrix of (1.3). Let ¯A:=ID+DA=[ˉaij] and ¯A:=ID+DA=¯B++¯C, where

    ¯B+=[ˉa11ˉr+1ˉa1nˉr+1ˉan1ˉr+nˉannˉr+n],¯C=[ˉr+1ˉr+1ˉr+nˉr+n],

    and ˉr+i:=max{0,ˉaij|ji}.

    Note that

    ˉaij={1di+diaii,i=j,diaij,ij.

    It follows that

    ˉr+i:=max{0,ˉaij|ji}=max{0,diaij|ji}=dimax{0,aij|ji}=dir+i, (2.1)

    and

    ˉaijˉr+i={1di+di(aiir+i)i=j,di(aijr+i),ij. (2.2)

    Since A=B++C, it holds that

    ¯A=ID+DA=ID+D(B++C)=(ID+DB+)+DC.

    Note that B+=[bij] and bij=aijr+i. Hence, from (2.1) and (2.2), we easily obtain that

    ¯B+=ID+DB+and¯C=DC.

    Let ¯B+=[ˉbij]. Then,

    ˉbij={1di+dibii,i=j,dibij,ij.

    Since A is a CKV-type B-matrix, then B+=[bij] is a CKV-type matrix with positive diagonal entries. Thus, by Lemma 2.3, it follows that for each iN, there exists SSi(B+), which implies that

    {|bii|>rSi(B+),(|bii|rSi(B+))(|bjj|r¯Sj(B+))>r¯Si(B+)rSj(B+)forallj¯S. (2.3)

    Hence, for each iN, it follows from (2.3) that

    ˉbiirSi(¯B+)=1di+di(biirSi(B+))>0,

    and that for all j¯S, if di0 and dj0, then

    (|ˉbii|rSi(¯B+))(|ˉbjj|r¯Sj(¯B+))=[1di+di(biirSi(B+))][1dj+dj(bjjr¯Sj(B+))]di(biirSi(B+))dj(bjjr¯Sj(B+))>dir¯Si(B+)djrSj(B+)=r¯Si(¯B+)rSj(¯B+),

    and if di=0 or dj=0, then

    (|ˉbii|rSi(¯B+))(|ˉbjj|r¯Sj(¯B+))=[1di+di(biirSi(B+))][1dj+dj(bjjr¯Sj(B+))]>0=r¯Si(¯B+)rSj(¯B+).

    These mean that SSi(¯B+) for each iN. Therefore, from Definition 1.3, ¯B+ is a CKV-type matrix with positive diagonal entries, and consequently, ¯A=ID+DA is a CKV-type B-matrix from Definition 2.1.

    In the following, we give an infinity norm bound for the inverse of CKV-type B-matrices. First, two lemmas are listed.

    Lemma 2.4. [5] Let A=[aij]Cn×n,n2, be a CKV-type matrix. Then

    ||A1||maxiNminSSi(A)maxj¯SβSij(A),

    where Si(A) is given by Definition 1.3, and

    βSij(A)=|ajj|r¯Sj(A)+r¯Si(A)(|aii|rSi(A))(|ajj|r¯Sj(A))r¯Si(A)rSj(A).

    Lemma 2.5. [11] Suppose P=(p1,,pn)Te, where e=(1,,1) and pi0 for all iN, then

    ||(I+P)1||n1.

    Theorem 2.1. Let A=[aij]Rn×n,n2, be a CKV-type B-matrix, and B+=[bij] be the matrix of (1.3). Then

    ||A1||(n1)maxiNminSSi(B+)maxj¯SβSij(B+),

    where Si(B+) is defined as in Definition 1.3, and

    βSij(B+)=|bjj|r¯Sj(B+)+r¯Si(B+)(|bii|rSi(B+))(|bjj|r¯Sj(B+))r¯Si(B+)rSj(B+).

    Proof. Since A is a CKV-type B-matrix, so B+ is a CKV-type matrix with positive diagonal entries and also a Z-matrix. By Corollary 4 of [31], we know that B+ is an M-matrix and thus (B+)1 is nonnegative. Hence, from A=B++C in which B+ and C are given by (1.3), we have

    A1=(B+(I+(B+)1C))1=(I+(B+)1C)1(B+)1,

    which implies that

    ||A1||||(I+(B+)1C)1||||(B+)1||. (2.4)

    Note that C=(r+1,,r+n)Te is nonnegative. Therefore, (B+)1C can be written as (p1,,pn)Te, where pi0 for all iN. By Lemma 2.5, we get

    ||(I+(B+)1C)1||n1. (2.5)

    Since B+ is a CKV-type matrix, it follows from Lemma 2.4 that

    ||(B+)1||maxiNminSSi(B+)maxj¯SβSij(B+). (2.6)

    Hence, from (2.4), (2.5), and (2.6), the conclusion follows.

    Based on Theorem 2.1, we give in this section an upper bound of maxd[0,1]n||(ID+DA)1|| when A is a CKV-type B-matrix, and give some comparisons with other results. Before that, a useful lemma is needed.

    Lemma 3.1. [20,Lemma 3] Let γ>0 and η0. Then for any x [0,1],

    11x+γx1min{γ,1}

    and

    ηx1x+γxηγ.

    Theorem 3.1. Let A=[aij]Rn×n,n2, be a CKV-type B-matrix, and B+=[bij] be the matrix of (1.3). Then

    maxd[0,1]n||(ID+DA)1||(n1)maxiNminSSi(B+)maxj¯SαSij(B+), (3.1)

    where Si(B+) is defined as in Definition 1.3, and

    αSij(B+)=(biirSi(B+))(bjjr¯Sj(B+))max{1biirSi(B+),1}+(bjjr¯Sj(B+))r¯Si(B+)max{1bjjr¯Sj(B+),1}(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+).

    Proof. Since A is a CKV-type B-matrix, by Proposition 2.2, it follows that ID+DA is also a CKV-type B-matrix. Taking into account that A=B++C in which B+ and C are defined as (1.3), then

    ID+DA=ID+D(B++C)=ID+DB++DC.

    Denote ¯B+:=ID+DB+=[¯bij] and ¯C=DC. Then, from Theorem 2.1, we have

    ||(ID+DA)1||(n1)maxiNminSSi(¯B+)maxj¯SβSij(¯B+). (3.2)

    By Lemma 3.1, it follows that for all iN,j¯S,

    βSij(¯B+)=|ˉbjj|r¯Sj(¯B+)+r¯Si(¯B+)(|ˉbii|rSi(¯B+))(|ˉbjj|r¯Sj(¯B+))r¯Si(¯B+)rSj(¯B+)=1dj+dj(bjjr¯Sj(B+))+dir¯Si(B+)[1di+di(biirSi(B+))][1dj+dj(bjjr¯Sj(B+))]dir¯Si(B+)djrSj(B+)=11di+di(biirSi(B+))+dir¯Si(B+)[1di+di(biirSi(B+))][1dj+dj(bjjr¯Sj(B+))]1dir¯Si(B+)djrSj(B+)[1di+di(biirSi(B+))][1dj+dj(bjjr¯Sj(B+))]max{1biirSi(B+),1}+max{1bjjr¯Sj(B+),1}r¯Si(B+)biirSi(B+)1r¯Si(B+)biirSi(B+)rSj(B+)bjjr¯Sj(B+)=(biirSi(B+))(bjjr¯Sj(B+))max{1biirSi(B+),1}+(bjjr¯Sj(B+))r¯Si(B+)max{1bjjr¯Sj(B+),1}(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)=:αSij(B+).

    Furthermore, by the proof of Proposition 2.2, Si(B+)Si(¯B+) for each iN. Thus,

    maxiNminSSi(¯B+)maxj¯SβSij(¯B+)maxiNminSSi(B+)maxj¯SαSij(B+). (3.3)

    Hence, the conclusion follows from (3.2) and (3.3).

    Remark 3.1. Note that, if biirSi(B+)1 and bjjr¯Sj(B+)1, then

    αSij(B+)=bjjr¯Sj(B+)+r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+);

    If biirSi(B+)>1 and bjjr¯Sj(B+)1, then

    αSij(B+)=(biirSi(B+))(bjjr¯Sj(B+))+r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+);

    If biirSi(B+)1 and bjjr¯Sj(B+)>1, then

    αSij(B+)=bjjr¯Sj(B+)+(bjjr¯Sj(B+))r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+);

    If biirSi(B+)>1 and bjjr¯Sj(B+)>1, then

    αSij(B+)=(biirSi(B+))(bjjr¯Sj(B+))+(bjjr¯Sj(B+))r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+).

    Since a DZ-type-B-matrix is a CKV-type B-matrix, the bound (3.1) can also be used to estimate maxd[0,1]n||(ID+DA)1|| when A is a DZ-type-B-matrix. The following theorem provides that the bound (3.1) is better than the bound (1.4) in Theorem 1.1 (Theorem 6 of [24]).

    Theorem 3.2. Let A=[aij]Rn×n be a DZ-type-B-matrix, and B+=[bij] be the matrix of (1.3). Then (3.1) holds. Furthermore,

    maxiNminSSi(B+)maxj¯SαSij(B+)maxiNminjγi(B+)ζij(B+),

    where γi(B+) and ζij(B+) are given by Theorem 1.1, Si(B+) and αSij(B+) are defined in Theorem 3.1.

    Proof. For each iN, note that

    γi(B+):={jN{i}:(|bii|rji(B+))|bjj|>|bij|rj(B+)},

    and

    Si(B+):={SΣ(i):|bii|>rSi(B+),andforallj¯S,(|bii|rSi(B+))(|bjj|r¯Sj(B+))>r¯Si(B+)rSj(B+)}

    with Σ(i):={SN:iS}. Take S=N{j}, ¯S={j}, ji, then

    rSi(B+)=rji(B+),r¯Si(B+)=|bij|,rSj(B+)=rj(B+),r¯Sj(B+)=0,

    which leads to

    αSij(B+)=(biirSi(B+))(bjjr¯Sj(B+))max{1biirSi(B+),1}+(bjjr¯Sj(B+))r¯Si(B+)max{1bjjr¯Sj(B+),1}(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)=(biirji(B+))bjjmax{1biirji(B+),1}+bjj|bij|max{1bjj,1}(biirji(B+))bjj|bij|rj(B+)=ζij(B+).

    It is easy to see that jγi(B+) is equivalent to S=N{j}Si(B+). Therefore, for each iN,

    minSSi(B+)maxj¯SαSij(B+)=min{minS=N{j}Si(B+)maxj¯SαSij(B+),minSSi(B+)(N{j})maxj¯SαSij(B+)}=min{minjγi(B+)ζij(B+),minSSi(B+)(N{j})maxj¯SαSij(B+)}minjγi(B+)ζij(B+).

    This completes the proof.

    Particularly, for B-matrices, as an important subclass of CKV-type B-matrices, we next show that the bound (3.1) is better than that given by García-Esnaola and Peña in [10] in some cases.

    Theorem 3.3. [10,Theorem 2.3] Let ARn×n be a B-matrix, and B+=[bij] be the matrix of (1.3). Let βi:=biiri(B+) and β:=miniN{βi}. Then

    maxd[0,1]n||(ID+DA)1||(n1)1min{β,1}. (3.4)

    Theorem 3.4. Let A=[aij]Rn×n be a B-matrix, and B+=[bij] be the matrix of (1.3). Let βi:=biiri(B+), β:=miniN{βi}, and S be a nonempty proper subset of N. Then (3.1) holds. Furthermore, if biirSi(B+)1 for all iN and bjjr¯Sj(B+)1 for all j¯S, then,

    maxiNminSSi(B+)maxj¯SαSij(B+)1min{β,1}, (3.5)

    and if βi>1 for each iN, then

    maxiNminSSi(B+)maxj¯SαSij(B+)1min{β,1}, (3.6)

    where αSij(B+) is defined as in Theorem 3.1.

    Proof. By the fact that a B-matrix is a CKV-type B-matrix, we know that (3.1) holds directly. We now prove that (3.5) and (3.6) hold. For each iN, SSi(B+), and j¯S, if biirSi(B+)1 and bjjr¯Sj(B+)1, then from Remark 3.1 that

    αSij(B+)=bjjr¯Sj(B+)+r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+).

    If bjjrj(B+)<biiri(B+), then

    (bjjr¯Sj(B+))rSj(B+)<(biirSi(B+))r¯Si(B+),

    which implies that

    (bjjr¯Sj(B+))2(bjjr¯Sj(B+))rSj(B+)+r¯Si(B+)(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)<(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+),

    i.e.

    [(bjjr¯Sj(B+))rSj(B+)][(bjjr¯Sj(B+))+r¯Si(B+)]<(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+).

    It follows that

    (bjjr¯Sj(B+))+r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)<1(bjjr¯Sj(B+))rSj(B+)=1bjjrj(B+)=max{1biiri(B+),1bjjrj(B+)}1min{β,1}. (3.7)

    If bjjrj(B+)biiri(B+), then

    (bjjr¯Sj(B+))rSj(B+)(biirSi(B+))r¯Si(B+),

    implying that

    (biirSi(B+))(bjjr¯Sj(B+))(bjjr¯Sj(B+))r¯Si(B+)+(biirSi(B+))r¯Si(B+)(r¯Si(B+))2(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+),

    i.e.

    [(biirSi(B+))r¯Si(B+)][(bjjr¯Sj(B+))+r¯Si(B+)](biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+).

    It holds that

    (bjjr¯Sj(B+))+r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)1(biirSi(B+))r¯Si(B+)=1biiri(B+)=max{1biiri(B+),1bjjrj(B+)}1min{β,1}. (3.8)

    Hence, (3.5) follows from (3.7) and (3.8).

    If βi:=biiri(B+)>1 for each iN, then

    biirSi(B+)biiri(B+)>1andbjjr¯Sj(B+)bjjrj(B+)>1.

    By Remark 3.1, we can see that

    αSij(B+)=(biirSi(B+))(bjjr¯Sj(B+))+(bjjr¯Sj(B+))r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)(biirSi(B+))(bjjr¯Sj(B+))+rSj(B+)r¯Si(B+)(biirSi(B+))(bjjr¯Sj(B+))r¯Si(B+)rSj(B+)1.

    Therefore,

    maxiNminSSi(B+)maxj¯SαSij(B+)1=1min{β,1}.

    The proof is complete.

    Remark from Theorem 3.4 that we can take the minimum of bounds (3.1) and (3.4) to estimate the error bound for the LCP(A,q) with A being a B-matrix, that is,

    maxd[0,1]n||(ID+DA)1||(n1)min{1min{β,1},maxiNminSSi(B+)maxj¯SαSij(B+)}.

    In this section, three examples are given to show the advantage of the bound (3.1) in Theorem 3.1.

    Example 4.1. Consider the following matrix

    A=[4022032211621126].

    Obviously, B+=A and C=0. It is easy to verify that B+ is not a DZ-type matrix and an S-SDD matrix, consequently, not a SDD matrix and a DSDD matrix. Hence, A is not a DZ-type-B-matrix and an SB-matrix, and thus not a B-matrix and a DB-matrix. So we cannot use the error bounds in [6,7,8,10,20,24] to estimate maxd[0,1]4||(ID+DA)1||. However, by calculations, one has that B+ is a CKV-type matrix with positive diagonal entries, and thus A is a CKV-type B-matrix. So by the bound (3.1) in Theorem 3.1, we get

    maxd[0,1]4||(ID+DA)1||21.

    Example 4.2. Consider the following matrix

    A=[3022032211601106].

    Note that r+i:=max{0,aij|ji}=0 for i=1,2,3,4. Hence, B+=A and C=0. By calculations, we have that A is a DZ-type-B-matrix, and thus it is a CKV-type B-matrix. By the bound (3.1) in Theorem 3.1, we have

    maxd[0,1]4||(ID+DA)1||12.6,

    while by the bound (1.4) in Theorem 1.1, it holds that

    maxd[0,1]4||(ID+DA)1||27.

    Obviously, the bound (3.1) is sharper than bound (1.4) in Theorem 1.1 (Theorem 6 of [24]).

    Example 4.3. Consider the B-matrix

    A=[3111213112113120001].

    Note that B+=A, C=0. Then, by the bound (3.4) in Theorem 3.3, we have

    maxd[0,1]4||(ID+DA)1||6.

    In addition, A is also a CKV-type B-matrix. By calculations, for i=1,2,3, take S={1,2,3} and ¯S={4}, it follows that biirSi(B+)1 for all i{1,2,3,4} and b44r¯S4(B+)1; and for i=4, take S={4} and ¯S={1,2,3}, it follows that biirSi(B+)1 for all i{1,2,3,4} and bjjr¯Sj(B+)1 for all j¯S, which satisfy the hypothesis of Theorem 3.4. Therefore, by Theorem 3.4, we get

    maxd[0,1]4||(ID+DA)1||4.5,

    which is smaller than the bound (3.4) in Theorem 3.3 (Theorem 2.3 of [10]).

    In this paper, on the basis of the class of CKV-type matrices, a new subclass of P-matrices: CKV-type B-matrices, containing B-matrices, DB-matrices, SB-matrices as well as DZ-type-B-matrices, is introduced, and an upper bound for the infinity norm for the inverse of CKV-type B-matrices is provided. Then, by this bound, an error bound for the corresponding LCP(A,q) is given. We also proved that the new error bound is sharper than those of [10] and [24] in some cases, and give numerical examples to show the advantage of our results.

    The authors sincerely thank the editors and anonymous reviewers for their valuable suggestions and helpful comments, which greatly improved the quality of the paper. This work was supported by National Natural Science Foundation of China (31600299), Young Talent fund of University Association for Science and Technology in Shaanxi, China (20160234), the Natural Science Foundations of Shaanxi province, China (2017JQ3020, 2020JM-622), the Scientific Research Program Funded by Shaanxi Provincial Education Department (18JK0044), and the Key Project of Baoji University of Arts and Sciences (ZK2017021).

    The authors declare no conflict of interest.



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