Construction and decomposition of scaling matrices for Sk-SDD matrices and their application to linear complementarity problems

We introduce a novel subclass of H-matrices called S-k-strictly diagonally dominant(S-k-SDD) matrices, where k is any positive integer. These matrices generalize SDD matrices, S-SDD matrices, and generalized SDD1 matrices. We provide a method for constructing scaling matrices for S-k-SDD matrices, ensuring that their multiplication with the scaling matrix results in an SDD matrix. By decomposing the scaling matrix into a product of two matrices, we establish an upper bound on the infinity norm of the inverse matrix for S-k-SDD matrices. Moreover, based on the decomposition of the scaling matrix, we derive an error bound for the linear complementarity problem associated with S-k-SDD matrices. Significantly, our error bound represents a theoretical enhancement over the findings reported by P.F. Dai, Y.T. Li, and C.J. Lu in their paper titled "Error bounds for linear complementarity problems for SB-matrices" (Numerical Algorithms, 61(1): 121-139, 2012). Furthermore, we substantiate the effectiveness and superiority of our findings through numerical experiments conducted with randomly generated matrices.