New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

Abstract A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

In this paper, we give a new error bound for linear complementarity problems when the involved matrices are Σ-SDD matrices, which is dependent only on the entries of the involved matrix. As an application, we provide a new error bound for linear complementarity problem with SB-matrices. Numerical examples are reported to show that the obtained bounds are better than those in [17], [18] and [27] in some cases.

New error bounds for LCPs of Σ-SDD matrices
Let us rst introduce some basic notations. A matrix M = [m ij ] ∈ R n×n is a Z-matrix if all its o -diagonal entries are nonpositive, and a nonsingular M-matrix if M is a Z-matrix with M − being nonnegative [1]. Let N := { , . . . , n} and S denotes a proper nonempty subset of N, S := N \ S denotes its complement in N. For a given matrix M = [m ij ] ∈ C n×n , denote Remark here that Σ-SDD matrices were usually called S-strictly diagonally dominant matrices in [28].
In [27], García-Esnaola and Peña provide the following error bound for the linear complementarity problem involved with Σ-SDD matrices.
and if γ > , then where Recently, Wang et al. [29] proved that the in mum of error bounds (3) and (4) as a function of the parameter γ exists, and also can be determined. where and γ |P S |+ := min .
Observe from Theorem 2.4 and Theorem 2.5 that if A is a large matrix, then the calculations of P S andP S (or P S andP S ) in bounds (5) and (6) will be very complicated. On the other hand, for strictly diagonally dominant matrices, the bounds (5) and (6) become invalid. So it is interesting to nd alternative error bounds depending only on the elements of the matrices for the LCP(M, q) when M is a Σ-SDD matrix. We next address this problem, before that some lemmas are listed. .

Lemma 2.8. Let S be a nonempty proper subset of N, and M
it follows that for each i ∈ N, and By (7) and (8), we have that for each i ∈ S, and Similarly, for each j ∈ S,m and Hence, If d k = for some k ∈ N, then from (9) and (10) we get If < d k ≤ for some k ∈ N, then from the fact that M is a Σ-SDD matrix we obtain Now the conclusion follows from (9), (10), (11) and (12).
By Lemma 2.6, Lemma 2.7, and Lemma 2.8, we establish the rst main result of this paper. where where .
Note thatm Then by Lemma 2.6 and Lemma 2.8, it follows that for each i ∈ S, and Analogously, for each j ∈ S, we have and By (15), (16), (17) and (18), it follows that for each i ∈ S, j ∈ S, In a similar way, we can prove that for each i ∈ S, j ∈ S, The conclusion follows from (14), (19) and (20).

Remark 2.10.
Observe that bound (13) in Theorem 2.9 only depends on the elements of M, and it is easy to implement. For a set S with nite elements, we use |S| to denote the number of elements in the set S. From bound (13), we obtain the number of the basic arithmetic operations of bound (13) is |S| · |S| · ( n + ) (requiring |S| · |S| · [ (n − ) + ] additions and · |S| · |S| comparisons, multiplications and divisions of numbers). Furthermore, it follows from |S| < n and |S| < n that |S| · |S| · ( n + ) < n ( n + ). Thus, the bound (13) of Theorem 2.9 can be performed in polynomial time.
By Theorem 2.9, we can easily obtain the following result.
Next, three examples are given to show the advantage of the bound (13) in Theorem 2.9. Before that, a wellknown result which will be used later is given.  [14], where

Example 2.13. Consider the family of SDD matrices in
< .
Since a Σ-SDD matrix is an S-Nekrasov matrix, the bound (2.14) of Theorem 2.2 in [8] for S-Nekrasov matrices can also be used to estimate max shows that the bound (13) given in Theorem 2.9 is sharper than the bound (2.14) of Theorem 2.2 in [8].
Example 2.14. The LCP(M, q) has often been used to discuss formulation and solution of tra c equilibrium problems [32,33]. Consider the matrix M ∈ R × arising from a simple tra c network problem [32]: In contrast, bound (2.14) of Theorem 2.2 in [8] for S-Nekrasov matrices gives the following estimation:  It should be pointed out that the bound (13)

New error bounds for LCPs of SB-matrices
Based on Theorem 2.9, we in this section present a new error bound for linear complementarity problems associated with SB-matrices. For a real matrix M = [m ij ] ∈ R n×n , we can write it as where Obviously, B is a Z-matrix and C is a nonnegative matrix of rank . Let us recall the de nition of SB-matrices which is proposed by Li et al. in [34] as a subclass of P-matrices. (21)

De nition 3.1. A real matrix M = [m ij ] ∈ R n×n is called an S-strictly dominant B-matrix (SB-matrix) if it can be written in form
where Since M is an SB-matrix, then we can write M = B + C as in (21) Next, we give an upper bound for ||(B D ) − ||∞. Since B is a Σ-SDD matrix, we have from Theorem 2.9 that The conclusion follows from (23) and (24). which is drawn in Figure 4. It can be seen from Figure 4 that the bound (22) in Theorem 3.2 is smaller than the bound (26) in Theorem 3.5 (Theorem 2.4 in [18]). The bound (26) in Theorem 3.5 The bound (22) in Theorem 3.2

Conclusions
In this paper, for the linear complementarity problems with a Σ-SDD matrix M, we rst give an alternative error bound for the LCP(M, q) which depends only on the entries of M. Then, by this new result, a new error bound for the LCP(M, q) with SB-matrices is provided. We also illustrate the results by numerical examples, where we improve bounds obtained in [17] and [18].