Parallel Computer Algebra 1

2 Building and implementing parallel algorithms in the area of computer algebra has become an important thread of research for more than a decade with the increasing availability of various parallel architectures, from dedicated machines to network of workstations. New algorithms have been built and implemented to solve high performance computing challenges. The aim of this tutorial is to give an introduction to parallel algorithms in computer algebra, from the building of an efficient algorithm to its effective implementation on a given architecture. Parallel computer algebra systems, that exploit the parallelism of an algorithm on a given architecture, play a central role to ensure efficient executions. Due to the variety of parallel programming models, several such systems propose various approaches to express parallelism, from data distribution to functional parallelism. After an introduction to algorithmic techniques and classical programming models, the tuto-rial will focus on parallel computer algebra systems, parallel linear algebra algorithms and their effective implementations. The tutorial is organized in four parts : 1. Parallel efficient algorithms. The major techniques used to build efficient algorithms on theoretical machine models are presented. They are illustrated by various basic computer algebra algorithms. Due to the non-uniformity of memory access, communication complexity is a key point to take into account in the analysis of the algorithm. 2. Programming models and scheduling. To combine expressive power and portability, several programming models have been proposed, from message-passing to bulk-synchronous programming and functional languages. The inherent overhead due to their emulation makes each of them suited to a specific range of applications. 3. Parallel computer algebra systems. Different parallel systems are proposed that are based on the coupling of a sequential system and a parallel programming model. They are often guided by the classes of applications on which they have been experimented. 4. Parallel linear algebra. The parallelization techniques introduced before are illustrated on various research problems in parallel linear algebra : system solving, gcd, rank and normal forms.