  Step-asynchronous successive overrelaxation updates the values contained in a
single vector using the usual Gau\ss-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 $\sigma\geq\rho(A)$ and a vector $\boldsymbol
w>0$ such that $A\boldsymbol w\leq\sigma\boldsymbol w$, 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
$A\boldsymbol x$ is feasible, but estimating the supremum norm of $(sI-A)^{-1}$
is not.
