On absolute value equations associated with M-matrices and H-matrices

Abstract

We consider the absolute value equation (AVE) $Ax-\left|x\right|=b$, where the diagonal entries of $A\in R^{n\times n}$ are all greater than 1 and $〈A〉-I$ is an irreducible singular $M$-matrix ($〈A〉$ is the comparison matrix of $A$). We investigate the existence and uniqueness of solutions for the AVE. The AVE does not necessarily have a unique solution for every $b\in R^n$, so most of the existing convergence results for various iterative methods are not generally applicable. Moreover, the generalized Newton method may break down. We show that if the AVE has a solution $x^{\ast }$ with at least one negative component, then the sequence generated by the generalized Gauss–Seidel iteration converges to $x^{\ast }$ linearly for any initial vector.