On the convergence of the new SOR-like method for solving absolute value equations

This paper revisits the convergence of the new SOR-like (NSOR) method proposed by Dong et al. for solving the absolute value equation Ax - |x| = b. The existing analysis uses the fact that the spectral radius of the iteration matrix at each step is less than one as a convergence argument. However, this matrix depends on the sign of the current iterate, so this argument cannot guarantee convergence of the whole iteration process. To show this, we construct a one-dimensional counterexample in which the parameters satisfy the conditions in the original paper and the spectral radius of each iteration matrix is less than one, but the generated sequence still diverges. We then estimate successive iteration differences and build a fixed nonnegative control matrix W. Using the Cauchy convergence criterion, we prove that if rho(W) < 1, then the NSOR method converges from any initial point to the unique solution of the absolute value equation. Based on this result, we give a sufficient convergence region for the parameters and obtain the fixed pair (omega, sigma) = (1, 1) by minimizing the spectral radius of W. Numerical experiments show that the original claimed convergence region contains parameter points at which the iteration fails to reach the prescribed tolerance, and that the original choice of optimal parameters can lead to divergence in some examples. In contrast, the fixed pair (1, 1) may require more iteration steps, but gives lower total computing time in the experiments.