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.

翻译：我们考虑绝对值方程(AVE)$Ax-|x|=b$的广义牛顿法(GNM)。该方法只要收敛就具有有限终止性质，无论该绝对值方程是否存在唯一解。我们证明了当$\rho(|A^{-1}|)<1/3$时广义牛顿法收敛。针对$A-I$为非奇异$M$-矩阵或不可约奇异$M$-矩阵的情形，我们还给出了新的结果。当$A-I$为不可约奇异$M$-矩阵时，绝对值方程可能具有无穷多个解。在此情形下，我们证明了只要初始猜测至少含有一个非正分量，广义牛顿法必定终止于唯一可识别的解。