Arithmetic Stability of an Implementation

Abstract

Round-off errors are tied to the implementation, i.e. two different implementations of the same algorithm might exhibit different error propagation patterns. We introduce the term arithmetic stability and formalise how to find out if an implementation is stable under round-off errors. The formalism allows us to show that our previously studied N-body problem is indeed single-step stable and also stable over longer simulation runs. For implementations of extreme-scale scalar products, e.g., stability however does not come for free, and we have to carefully arrange all computational steps.