A technique to expedite iterative computations which is based on multidimensional chains of recurrences (MCR) is presented. Algorithms for MCR construction, interpretation and MCR-based code generation are discussed. The notion of delayed MCR simplification introduced here for the first time often leads to reduced times for both the MCR construction and MCR interpretation phases of this technique. Three different implementations of the MCR technique (in Maple, C and Java) are described. 1 Introduction We consider the problem of expediting computational tasks like this: given a closed form function G(x1 ; : : : ; xm ), initial points x01 ; : : : ; x0m and steps h1 ; : : : ; hm , compute values G(x1 ; : : : ; xm) for x1 = x01 ; x01 + h1 ; : : : ; x01 + n1h1 ; : : : ; xm = x0m ; x0m+hm ; : : : ; x0m+nmhm : Many problems appearing in such applications as plotting (explicit, parametric, implicit) in different coordinate systems, animation, signal processing etc. can be reduced to this kin...

INTRODUCTION Computer scientists should have a knowledge of abstract statistical thermodynamics. First, computer systems are dynamical systems, much like physical systems, and therefore an important first step in their characterization is in finding properties and parameters that are constant over time (i.e., constants of motion). Second, statistical thermodynamics successfully reduces macroscopic properties of a system to the statistical behavior of large numbers of microscopic processes. As computer systems become large assemblages of small components, an explanation of their macroscopic behavior may also be obtained as the aggregate statistical behavior of its component parts. If not, the elegance of the statistical thermodynamical approach can at least provide inspiration for new classes of models. 1 Third, the components of computer systems are approaching the same size as the microscopic pr

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...

The goal of this work was to answer one simple question: given that the trigonometric functions take hundreds of clock cycles to execute on a Pentium IV, can they be computed faster, especially given that all Intel processors now have fast floating-point hardware? The streaming SIMD extensions (SSE/SSE2) in every Pentium III and IV provide both scalar and vector modes of computation, so it has been our goal to use the vector hardware to compute the cosine and other trigonometric functions. The cosine function was chosen, as it has significant use in our research as well as in image construction with the discrete cosine transform.

. We report on a large-scale statistical evaluation of pseudorandom properties of certain cryptographic functions such as des and md5. The evaluation is based on the well-known birthday attack. The attack requires large amounts of memory. We describe a parallel algorithm which can exploit the large amounts of secondary memory (local disks) available on many workstation clusters and parallel machines. The overheads due to communication and disk accesses can be minimized by techniques similar to those used in parallel data bases for parallel external sorting. We have implemented the algorithm using the message passing interface MPI. We display performance measurements on an IBM SP2 which show that the costs for communication and disk accesses are negligible. 1 Introduction Cryptography plays an important role in many fields. For example, applications such as electronic banking, pay-TV, and business over the internet depend on the availability of secure cryptographic functions. Often, ...