Tight Interval Inclusions with Compensated Algorithms

Abstract

Compensated algorithms consist in computing the rounding errors of individual operations and then adding them later on to the computed result. This makes it possible to increase the accuracy of the computed result efficiently. Computing the rounding error of an individual operation is possible through the use of a so-called error-free transformation. In this article, we show that it is possible to use compensated algorithms for having tight interval inclusions. We study compensated algorithms for summation, dot product and polynomial evaluation. We prove that the use of directed rounding makes it possible to get narrow inclusions with compensated algorithms. This is due to the fact that error-free transformations are no more exact but still sufficiently accurate to improve the numerical quality of results.