CONVERTING APPROXIMATE ERROR ROUNDS INTO EXACT ONES

In order to produce error bounds quickly and easily, people often appl y to error bounds linearized propagation rules. This is done instead o f 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 s uch approximate error bounds into true bounds. This is done by dividin g the approximate bound by 1 - delta, with a small delta. Both the app roximate bound and delta are produced by the same linearized error ana lysis. This method makes it possible both to simplify the error analys es and to sharpen the bounds in an interesting class of numerical algo rithms. In particular it seems to be ideal for the derivation of tight , true error bounds for simple and accurate algorithms, like those use d in subroutines for the evaluation of elementary mathematical functio ns (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 forwa rd a posteriori error analysis theory and of running error analysis ar e given also. In the course of proofs a new methodology is applied for the representation of propagated error bounds. This methodology promo tes easy derivation of sharp, helpful inequalities. Several examples o f forward a priori error analysis and one of a posteriori error analys is and running error analysis are included.