An improved error bound for linear complementarity problems for B-matrices

A new error bound for the linear complementarity problem when the matrix involved is a B-matrix is presented, which improves the corresponding result in (Li et al. in Electron. J. Linear Algebra 31(1):476-484, 2016). In addition some sufficient conditions such that the new bound is sharper than that in (García-Esnaola and Peña in Appl. Math. Lett. 22(7):1071-1075, 2009) are provided.


Introduction
Given an n × n real matrix M and q ∈ R n , the linear complementarity problem (LCP) is to find a vector x ∈ R n satisfying When the matrix M for the LCP(M, q) belongs to P-matrices or some subclass of Pmatrices, various bounds for () were proposed; e.g., see [ and r + i = max{, m ij |j = i}. Then It is not difficult to see that the bound () will be inaccurate when the matrix M has very small value of min i∈N {b iij =i |b ij |}; for details, see [, ]. To conquer this problem, Li et al., in [] gave the following bound for () when M is a B-matrix, which improves those provided by Li and Li in [, ].
In this paper, we further improve error bounds on the LCP(M, q) when M belongs to B-matrices. The rest of this paper is organized as follows: In Section  we present a new error bound for (), and then prove that this bound is better than those in Theorems  and . In Section , some numerical examples are given to illustrate our theoretical results obtained.

Main result
In this section, an upper bound for () is provided when M is a B-matrix. Firstly, some definitions, notation and lemmas which will be used later are given as follows.
A matrix A = [a ij ] ∈ C n,n is called a strictly diagonally dominant (SDD) matrix if |a ii | > n j =i |a ij | for all i = , , . . . , n. A matrix A = [a ij ] ∈ R n,n is called a nonsingular M-matrix if its inverse is nonnegative and all its off-diagonal entries are nonpositive []. In [] it was proved that a B-matrix has positive diagonal elements, and a real matrix A is a B-matrix if and only if it can be written in the form () with B + being a SDD matrix. Given a matrix A = [a ij ] ∈ C n,n , let Lemmas  and  will be used in the proofs of the following lemma and Theorem . Then Furthermore, it follows from (), () and Lemma  that for each i = j (j < i ≤ n) The proof is completed.
By Lemmas , ,  and , we give the following bound for () when M is a B-matrix. is the matrix of (). Then Next, we give an upper bound for (B + D ) - ∞ . By Lemma , we have where By Lemmas  and , we can easily see that, for each i ∈ N , and that, for each k ∈ N , Furthermore, according to Lemma  and (), it follows that, for each j ∈ N ,
The comparisons of the bounds in Theorems  and  are established as follows.
is the matrix of (). Letβ i and β i be defined in Theorems  and , respectively. Then Proof Note that Hence, for each i ∈ N This completes the proof.
Remark here that, whenβ i <  for all i ∈ N , then Next it is proved that the bound () given in Theorem  can improve the bound () in Theorem  (Theorem . in []) in some cases.
Proof When β >  and α <  β , we can easily get Similarly, for β <  and αβ < β, the conclusion can be proved directly.

Numerical examples
Two examples are given to show that the bound in Theorem  is sharper than those in Theorems  and .
By Theorem , we find that, for any k ≥ , where √ -  < a <  and -a  +a < k < . Then M k = B + k + C with C is the null matrix.
By simple computations, we can get It is not difficult to verify that M k satisfies the condition (i) of Theorem . Thus, the bound () of Theorem  (Theorem . in which is larger than the bound