Abstract
We present two methods to compute Gröbner bases in parallel, both based on Buchberger's sequential algorithm. A distributed memory MIMD computer (the FPS 140) gives experimental results obtained with boolean polynomials.
The algorithms were implemented on the FPS T40 connected as a ring and as a hypercube of processors. The first implementation shows the interest of the parallelization. The second one, based on a divide and conquer strategy, has a behavior very close to the sequential algorithm.
We evaluate the contribution of the parallelism by a direct comparison of sequential and parallel times without references to complexity.
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J. M. Aroca, H. Hironaka, J. L. Vincente, “The theory of Maximal Contact,” Memorias de Mathematica del Instituto “Jorge Juana,” 29, (1975).
B. Buchberger, “Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungsystems,” Aequationes Math, 4, pp. 374–383, (1970).
B. Buchberger, “A criterion for detecting unnecessary reductions in the construction of Gröbner bases,” Proc. EUROSAM 79 (Ed. W. Ng), Springer LNCS 72, pp. 3–21, (1979).
B. Buchberger, “Gröbner Bases: an algorithmic method in polynomial ideal theory,” Recent Trends in Multidimensional Systems Theory (Ed. N. K. Bose), Reidel, (1985).
H. Melenk, H. M. Möller, W. Neun, “On Gröbner Basis Computation on a Supercomputer Using Reduce,” FB Mathematik und Informatik der Fernuniversität Hagen, Preprint SC 88-2, (1988).
C. Ponder, “Parallelism Algorithms for Gröbner Basis Reduction,” Electrical Engineering and Computer Sciences Dept, University of California, Berkeley, CA, (1988).
P. Sénéchaud, “Bases de Gröbner Booléennes Méthodes de Calcul; Applications; Parallelisation,” Thèse INPG, (1990).
W. Trinks, “On B. Buchberger's Method for Solving Algebraic Equations,” J. Number Theory, 10/4, pp. 475–488, (1978).
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, Ph. D. Thesis, University of Waterloo, Waterloo, Ontario (1985).
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© 1992 Springer-Verlag Berlin Heidelberg
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Senechaud, P. (1992). Boolean Gröbner bases and their MIMD implementation. In: Zippel, R.E. (eds) Computer Algebra and Parallelism. CAP 1990. Lecture Notes in Computer Science, vol 584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55328-2_8
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DOI: https://doi.org/10.1007/3-540-55328-2_8
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