Towards efficient parallelizations of a computer algebra algorithm

Abstract

This paper summarizes the results of a preliminary study that examines the feasibility of implementing computer algebra systems on massively pamllel (SIMD) architectures. O n serial computers, these systems rely on Buchberger's algorithm t o compute Gr6bner bases. A pamllelixation of this algorithm is proposed, which addresses the potential growth in the number of polynomials that can be generated during the computation. The parallel algorithm was implemented on a Connection Machine CM-200 System. The experimental results which were obtained for seven test problems are examined, and the algorithm and its implementation are evaluated. The results of this study provide insights into ongoing research t o develop more eficient parallel algorithms for finding Grlibner bases. of algebra and geometry that must be solved computationally [7].B. Buchberger's work on Grobner bases [3, 41 inspired the development of algorithms to solve algebraic and geometric problems similar to those mentioned above. However, the actual computational complexity of these algorithms has hindered their use in largescale applications. For this reason, we have initiated a study to investigate the feasibility of computing Grobner bases on massively parallel architectures such as the MasPar or Connection Machine systems. The ability to do this would enable scientists to obtain solutions to a larger class of problems that contain a substantial number of polynomials and/or variables. Further, the introduction of parallelism could reduce the amount of computation time needed to obtain these solutions. This paper is a preliminary report that summarizes the current status of our study.

HPC‐GAP: engineering a 21st‐century high‐performance computer algebra system

“…Roch et al [44] discuss the implementation of the parallel computer algebra system PAC on the Floating Point System hypercube Tesseract 20 and study the performance of a parallel Gröbner Basis algorithm on up to 16 nodes. Another parallel Gröbner Basis algorithm is implemented on a Cray Y-MP by Neun and Melenek [45] and later on a Connection Machine by Loustaunau and Wang [46]. We are not aware of any other work within the last 20 years that targets HPC for computational algebra.…”

High-Performance Computer Algebra: A Hecke Algebra Case Study

“…Roch et al [24] discuss the implementation and performance of a parallel Gröbner basis algorithm on the Floating Point System hypercube Tesseract 20 with 16 nodes. Another parallel Gröbner basis algorithm is implemented on a Cray Y-MP by Neun and Melenek [23] and later on a Connection Machine by Loustaunau and Wang [19]. We are not aware of any other work within the last 20 years that targets HPC for computational algebra.…”