How Fast Can We Compute Products?

In this paper we consider the problem of fast computation of n-ary products, for large n, over arbitrary precision integer or rational number domains. The combination of loop unrolling, chains of recurrences techniques and analogs of binary powering allows us to obtain order-of-magnitude speed improvements for such computations. Three different implementations of the technique (in Maple, C++ and Java) are described. Many examples together with timings are given. 1 Introduction An arbitrary precision arithmetic is undoubtedly the "work horse" of general purpose computer algebra systems and numerous specialized packages. Advanced algorithms to perform basic operations on arbitrary precision integers are very well known. Many books [1, 4, 7] give overviews of those algorithms together with detailed implementation remarks. Most computer algebra systems (such as Maple [3]) and specialized number theory packages (such as NTL [8]) contain implementations of these algorithms. For example for..