Converting approximate error bounds into exact ones

Abstract:

In order to produce error bounds quickly and easily, people often apply to error bounds linearized propagation rules. This is done instead of a precise error analysis. The payoff: Estimates so produced are not guaranteed to be true bounds. One can at most hope that they are good approximations of true bounds. This paper discusses a way to convert such approximate error bounds into true bounds. This is done by dividing the approximate bound by $1 - \delta$, with a small $\delta$. Both the approximate bound and $\delta$ are produced by the same linearized error analysis. This method makes it possible both to simplify the error analyses and to sharpen the bounds in an interesting class of numerical algorithms. In particular it seems to be ideal for the derivation of tight, true error bounds for simple and accurate algorithms, like those used in subroutines for the evaluation of elementary mathematical functions (EXP, LOG, SIN, etc.), for instance. The main subject of this paper is forward a priori error analysis. However, the method may be fitted to other types of error analysis too. In fact the outlines of a forward a posteriori error analysis theory and of running error analysis are given also. In the course of proofs a new methodology is applied for the representation of propagated error bounds. This methodology promotes easy derivation of sharp, helpful inequalities. Several examples of forward a priori error analysis and one of a posteriori error analysis and running error analysis are included.