Using the range for the infinity norm of inverse matrix of a strictly diagonally dominant $M$-matrix, some new error bounds for the linear complementarity problem are obtained when the involved matrix is a $B^S$-matrix. Theory analysis and numerical examples show that these upper bounds are more accurate than some existing results.

In this paper, an error bound for linear complementarity problems of strong $SDD$ ${}_1$ matrices is given. By properties of $SDD$ ${}_1$ matrices, a new subclass of $P$-matrices called $SD{D}_1$- $B$ is presented, which contains $B$-matrices. A new error bound of linear complementarity problems for $SD{D}_1$- $B$ is also provided, which improves the corresponding results. Numerical examples are given to illustrate the effectiveness of the obtained results.