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 $sigmageqrho(A)$ and a vector $boldsymbol w>0$ such that $Aboldsymbol wleqsigmaboldsymbol w$, every iteration of step-asynchronous successive overrelaxation for the problem $(sI- A)boldsymbol x=boldsymbol b$, with $s >sigma$, reduces geometrically the $boldsymbol w$-norm of the current error by a factor that we can compute explicitly. Then, we show that given a $sigma>rho(A)$ 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 $Aboldsymbol x$ is feasible, but estimating the supremum norm of $(sI-A)^{-1}$ is not.
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