Exact solution of linear systems over rational numbers by parallel p-adic arithmetic
We describe a parallel implementation of an algorithm for solving systems of linear equations over the field of rational numbers based on Gaussian elimination. The rationals are represented by truncated p-adic expansion. This approach permits us to do error free computations directly over the rationals without converting the system to an equivalent one over the integers. The parallelization is based on a multiple homomorphic image technique and the result is recovered by a parallel version of the Chinese remainder algorithm. Using a MIMD machine, we compare the proposed implementation with the classical modular arithmetic, showing that truncated p-adic arithmetic is a feasible tool for solving systems of linear equations. The proposed implementation leads to a speedup up to seven by ten processors with respect to the sequential implementation.
Unable to display preview. Download preview PDF.
- 1.Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison Wesley Publishing Company, 1975.Google Scholar
- 2.Bruce W. Char, Gregory J. Fee, Keith O. Geddes, Gaston H. Gonnet, Michael B. Monagan, and Stephen M. Watt. A tutorial introduction to Maple. Journal of Symbolic Computation, Vol.2, n.2, 1986.Google Scholar
- 3.Attilio Colagrossi, Carla Limongelli, and Alfonso Miola. Scientific computation by error-free arithmetics. Journal of information Science and Technology, July–October 1993.Google Scholar
- 4.George E. Collins et al. A Saclib 1.1 user's guide. Technical report, RISC-Linz, Johannes Kepler University, Austria, 1993. RISC-Linz Report Series n. 93-19.Google Scholar
- 5.Robert T. Gregory and Edayathumangalam V. Krishnamurthy. Methods and Applications of Error-Free Computation. Springer Verlag, 1984.Google Scholar
- 6.Hoon Hong et al. A Paclib user manual. Technical report, RISC-Linz, Johannes Kepler University, Austria, 1992. RISC-Linz Report Series n. 92-32.Google Scholar
- 7.Carla Limongelli. The Integration of Symbolic and Numeric Computation by p-adic Construction Methods. PhD thesis, Universitá degli Studi di Roma ”La Sapienza”, Italy, 1993.Google Scholar
- 8.Carla Limongelli. On an efficient algorithm for big rational number computations by parallel p-adics. Journal of Symbolic Computation, Vol.15, n.2, 1993.Google Scholar
- 9.Maurice Mignotte. Some useful bounds. In B. Buchberger, G. E. Collins, and R. Loos, editors, Computer Algebra Symbolic and Algebraic Computation, pages 259–263. Springer Verlag, 1983.Google Scholar
- 10.Henryk Minc and Marvin Marcus. Introduction to Linear Algebra. Macmillan, New York, 1965.Google Scholar
- 11.Alfonso Miola. Algebraic approach to p-adic conversion of rational numbers. Information Processing Letters, 18:167–171, 1984.Google Scholar
- 12.Roberto Pirastu and Kurt Siegl. Parallel computation and indefinite summation: A ∥Maple∥ application for the rational case. submitted to Journal of Symbolic Computation, 1994.Google Scholar
- 13.Kurt Siegl. Parallelizing algorithms for symbolic computation using ∥MAPLE∥. In Fourth ACM SIGPLAN Symp. on Principles and Practice of Parallel Programming, San Diego, pages 179–186, 1993.Google Scholar