p-adic Arithmetic and Parallel Symbolic Computation: An Implementation for Solving Linear Systems Over Rationals

Abstract

In this work we describe the use of truncated p-adic expansion of handling rational numbers by parallel algorithms for symbolic computation. As a case study we propose a parallel implementation for solving linear systems over the rationals.
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 working directly over rational numbers. A safe algorithm for computing the p-adic division operation is proposed.
The implementation leads to a speedup of up to seven by ten processors with respect to the sequential implementation.