Comments on finite termination of the generalized Newton method for absolute value equations

We consider the generalized Newton method (GNM) for the absolute value equation (AVE) $Ax-\mathrm{\mid }x\mathrm{\mid }=b$. 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 $\rho (\mathrm{\mid }{A}^{-1}\mathrm{\mid })<1\mathrm{/}3$. 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.