<p>An extended matrix splitting (EMS) iteration method is developed to solve the generalized absolute value equation (GAVE). The new algorithm framework encompasses thirteen existing classical methods, including Newton-type methods, SOR-like iteration methods, fixed point iteration methods, as well as their variants. Notably, it is particularly compatible with relaxed-based matrix splitting (RMS) iteration method, which itself covers a wide variety of methods. In addition, the EMS iteration method naturally yields several relaxation versions, further enhancing its flexibility. Theoretically, without the hypothesis of solvability, we prove that the EMS iteration method globally converges to the unique solution of the GAVE. Consequently, some convergence results of the existing methods are improved. Furthermore, a comparison theorem is established to demonstrate that the EMS iteration method can accelerate the convergence rate of the RMS iteration method. Numerical results verify the feasibility and efficiency of the proposed method for solving GAVE, particularly in its applications on the linear complementarity problem and the ridge regression problem.</p>