The algebraic eigenvalue problem
The algebraic eigenvalue problem
The Organization of Computations for Uniform Recurrence Equations
Journal of the ACM (JACM)
An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations
Journal of the ACM (JACM)
The Art of Computer Programming Volumes 1-3 Boxed Set
The Art of Computer Programming Volumes 1-3 Boxed Set
The complexity of parallel evaluation of linear recurrence
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Feasible arithmetic computations: Valiant's hypothesis
Journal of Symbolic Computation
Parallelism and algorithms for algebraic manipulation: current work
ACM SIGSAM Bulletin
On the parallel solution of parabolic equations
ICS '89 Proceedings of the 3rd international conference on Supercomputing
The Complexity of Parallel Evaluation of Linear Recurrences
Journal of the ACM (JACM)
Throughput optimization of general non-linear computations
ICCAD '99 Proceedings of the 1999 IEEE/ACM international conference on Computer-aided design
Some remarks on parallel exponentiation (extended abstract)
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
A Survey of Parallel Machine Organization and Programming
ACM Computing Surveys (CSUR)
ACM Transactions on Programming Languages and Systems (TOPLAS)
Parallel Processing for Real-Time Simulation: A Case Study
IEEE Parallel & Distributed Technology: Systems & Technology
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The parallel evaluation of rational expressions is considered. New algorithms which minimize the number of multiplication or division steps are given. They are faster than the usual algorithms when multiplication or division takes more time than addition or subtraction. It is shown, for example, that xn can be evaluated in two steps of parallel division and ⌈log2 n⌉ steps of parallel addition, while the usual algorithm takes ⌈log2 n⌉ steps of parallel multiplication.Lower bounds on the time required are obtained in terms of the degree of the expressions to be evaluated. From these bounds, the algorithms presented in the paper are shown to be asymptotically optimal. Moreover, it is shown that by using parallelism the evaluation of any first-order rational recurrence of degree greater than 1, e.g. yi+1 = 1/2;(yi + a/yi), and any nonlinear polynomial recurrence can be sped up at most by a constant factor, no matter how many processors are used and how large the size of the problem is.