Supremum-Norm Convergence for Step-Asynchronous Successive Overrelaxation on M-matrices

Step-asynchronous successive overrelaxation updates the values contained in a single vector using the usual Gauß–Seidel-like weighted rule, but arbitrarily mixing old and new values, the only constraint being temporal coherence— you cannot use a value before it has been computed. We show that given a nonnegative real matrix A, a σ ≥ ρ(A) and a vector w > 0 such that Aw ≤ σw, every iteration of step-asynchronous successive overrelaxation for the problem (sI −A)x = b, with s > σ, reduces geometrically the w-norm of the current error by a factor that we can compute explicitly. Then, we show that given a σ > ρ(A) it is in principle always possible to compute such a w. This property makes it possible to estimate the supremum norm of the absolute error at each iteration without any additional hypothesis on A, even when A is so large that computing the product Ax is feasible, but estimating the supremum norm of (sI −A)−1 is not. Mathematical Subject Classification: 65F10 (Iterative methods for linear systems)