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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 $σ\geqρ(A)$ and a vector $\boldsymbol w>0$ such that $A\boldsymbol w\leqσ\boldsymbol w$, every iteration of step-asynchronous successive overrelaxation for the problem $(sI- A)\boldsymbol x=\boldsymbol b$, with $s >σ$, reduces geometrically the $\boldsymbol w$-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 $\boldsymbol w$. 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 $A\boldsymbol x$ is feasible, but estimating the supremum norm of $(sI-A)^{-1}$ is not.