A chained-matrices approach for parallel computation of continued fractions and its applications

A chained-matrices approach for parallel computing thenth convergent of continued fractions is presented. The resulting algorithm computes the entire prefix values of any continued fraction inO(logn) time on the EREW PRAM model or a network withO(n/logn) processors connected by the cube-connectedcycles, binary tree, perfect shuffle, or hypercube. It can be applied to approximate the transcendental numbers, such as ande, inO(logm) time by usingO(m/logm) processors for a result withm-digit precision. We also use it to costoptimally solve the second-order linear recurrence, the polynomial evaluation, the recurrence of vector norm, the general class of recurrence equation defined by Kogge and Stone (1973), and the generalmth order linear recurrence. It is easy to implement because there are only some matrix multiplications and a division operation involved.This work was supported in part by National Science Council of the Republic of China under Contract NSC 77-0408-E002-09.     

A Pinwheel Scheduler for Three Distinct Numbers with a Tight Schedulability Bound

Given a multiset of positive integers , the pinwheel problem is to find an infinite sequence over such that there is at least one symbol i within any subsequence of length a _i . The density of A is defined as . In this paper we limit ourselves to instances composed of three distinct integers. The best scheduler [5] published previously can schedule all instances with a density of less than 0.77. A new and fast scheduling scheme based on spectrum partitioning is presented in this paper which improves the 0.77 result to a new density threshold. This scheduler has achieved the tight schedulability bound of this problem. Received February 17, 1995; revised July 22, 1996.