We consider the generalized Newton method (GNM) for the absolute value equation (AVE) \(Ax-|x|=b\) . The method has finite termination property whenever it is convergent, no matter whether the AVE has a unique solution. We prove that GNM is convergent whenever \(\rho (|A^{-1}|)<1/3\) . We also present new results for the case where \(A-I\) is a nonsingular M-matrix or an irreducible singular M-matrix. When \(A-I\) is an irreducible singular M-matrix, the AVE may have infinitely many solutions. In this case, we show that GNM always terminates with a uniquely identifiable solution, as long as the initial guess has at least one nonpositive component.
