Many problems in different fields may lead to solutions of absolute value equations, such as linear programming problems, linear complementarity problems, quadratic programming, mixed integer programming, the bimatrix game and so on. In this paper, by introducing a nonnegative real parameter to the modified Newton-based iteration scheme, we present a new relaxed modified Newton-based (RMN) iteration method for solving generalized absolute value equations. The famous Picard iteration method and the modified Newton-type iteration method are the exceptional cases of the RMN iteration method. The convergence property of the new method is discussed. Finally, the validity and feasibility of the RMN iteration method are verified by experimental examples.

In this paper, a relaxed Newton-type matrix splitting (RNMS) iteration method is proposed for solving the generalized absolute value equations, which includes the Picard method, the modified Newton-type (MN) iteration method, the shift splitting modified Newton-type (SSMN) iteration method and the Newton-based matrix splitting (NMS) iteration method. We analyze the sufficient convergence conditions of the RNMS method. Lastly, the efficiency of the RNMS method is analyzed by numerical examples involving symmetric and non-symmetric matrices.