Two class of synchronous matrix multisplitting schemes for solving linear complementarity problems (Q550095)

The authors consider the following linear complementarity problem: \[ \text{Find a vector }x\in\mathbb{R}^n\text{ such that }Mx+ q\geq 0,\;x\geq 0, x^T(Mx+ q)= p,\tag{\(*\)} \] where \(M\) is a \((n,n)\)-matrix and \(q\in\mathbb{R}^n\). The generalized accelerated overrelaxation and the symmetric successive overrelaxation procedures are used to establish two synchronous matrix multisplitting methods to solve the problem \((*)\). Concepts of \(H\)-matrix and \(M\)-matrix are introduced and convergence results under the assumption that \(M\) is an \(H\)-matrix or an \(M\)-matrix, as well as conditions for the monotone convergence of the proposed methods are presented. Numerical results presented in the concluding part of the paper show that the proposed methods are effective for solving linear complementarity problems with large and sparse matrices.