Interval arithmetic applied to polynomial remainder sequences

Abstract

Polynomial remainder sequences are the basis of many important algorithms in symbolic and algebraic manipulation. In a number of these algorithms, the actual coefficients of the sequence are not required; rather, the method uses the signs of the coefficients. Present techniques, however, compute the exact coefficients (or a mixed radix representation of them), and then obtain the signs. This paper discusses a new approach in which interval arithmetic is used to obtain the signs of the coefficients without computing their exact values. Comparisons of this method with analogous standard techniques show empirical computing time reductions of two orders of magnitude for even relatively small cases.