Abstract
This article presents a system which computes Gröbner bases on a shared memory multiprocessor. The basic idea is that each processor picks an element in the set of unreduced critical pairs, reduces the S-polynomial associated with it and updates the basis and the set of pairs according to the result. The originality of this algorithm relies on the small amount of synchronization it requires among the processes. The details of an implementation on a 16 processors Encore machine are given together with results of tests performed with well-known examples of the literature.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
M. Accetta, R. Baron, D. Golub, R. Rashid, A. Tevanian, and M. Young. Mach: A New Kernel Foundation For UNIX Development. In Proceedings Summer 1986 USENIX Technical Conference and Exhibition, pages 93–112, June 1986.
W. Boege, R. Gebauer, and H. Kredel. Some Examples for Solving Systems of Algebraic Equations by Calculating Gröbner Bases. Journal of Symbolic Computation, 1:83–98, 1986.
P. Brinch Hansen. Operating System Principles. Prentice-Hall, 1973.
B. Buchberger. An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal. PhD thesis, Univ. of Innsbruck, 1965.
B. Buchberger, Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. In N.K. Bose, editor, Recent Trends in Multidimensional Systems Theory, pages 184–232, D.Reidel Publishing Company, 1985.
B. Buchberger. The Parallelization of Critical-Pair/Completion Procedures on the L-Machine. In Proceedings of the Japanese Symposium on Functional Programming, pages 54–61, February 1987.
B. Buchberger. User's Manual for GRÖBNER, a Gröbner Bases Package in muMATH. Technical Report RISC-Linz 87-38, University of Linz (Austria), 1987.
E.C. Cooper and R.P. Draves. C Threads. Technical Report CMU-CS-88-154, Computer Science Department, Carnegie Mellon University, June 1988
R. Gebauer and M. Möller. On an installation of Buchberger's algorithm. Journal of Symbolic Computation, 6(2 & 3):275–286, 1988
S. Owicki. Axiomatic Proof Techniques for Parallel Programs. PhD thesis, Cornell University, 1975.
C.G. Ponder. Evaluation of “Performance Enhancements” in Algebraic Manipulation Systems. Technical report UCB/CSD/ 88/438, Computer Science Division, University of California, Berkeley, August 1988.
L. Robbiano. Gröbner Bases: a foundation for Commutative Algebra. In Computers and Mathematics, 1989.
P. Senechaud. Implementation of a parallel algorithm to compute a Gröbner basis on Boolean polynomials. In Computer Algebra and Parallelism, pages 159–166, Academic Press, 1989.
C. Traverso and L. Donati. Experimenting the Gröbner basis algorithm with the AIPI system. In Proceedings ISSAC 1989, 1989.
J.P. Vidal. The computation of Gröbner bases on a shared memory multiprocessor. Technical Report, School of Computer Science, Carnegie Mellon University, to appear.
S. Watt. Bounded Parallelism in Computer Algebra. PhD thesis, University of Waterloo Ontario, 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Vidal, JP. (1990). The computation of Gröbner bases on a shared memory multiprocessor. In: Miola, A. (eds) Design and Implementation of Symbolic Computation Systems. DISCO 1990. Lecture Notes in Computer Science, vol 429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52531-9_127
Download citation
DOI: https://doi.org/10.1007/3-540-52531-9_127
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52531-8
Online ISBN: 978-3-540-47014-4
eBook Packages: Springer Book Archive