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

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