Generalized SOR-like Methods for Solving Absolute Value Equations

For solving the absolute value equations Ax-|x|=b, a generalized SOR-like (GSOR) method is proposed by introducing the preconditioning matrix and using the relaxation parameter matrix instead of a single relaxation parameter. With the appropriate preconditioning matrices or parameters, the GSOR method can reduce to an existing SOR-like (NSOR) method or lead to some new SOR-like methods. Moreover, based on the unique solvability of Ax-|x|=b, the convergence theory of the GSOR method is established and its quasi-optimal parameters are given. In particular, by using truncated Neumann expansion, a new preconditioning matrix is constructed and a special GSOR method(GSOR-1) is derived. It has been proved that the GSOR-1 method has the smaller convergence factor than the NSOR method. Numerical tests further reveal that the GSOR-1 method has faster convergence rate and costs less computational times than the NSOR method.