Parallel systems in symbolic and algebraic computation

This thesis describes techniques that exploit the distributed memory in massively parallel processors to satisfy the peak memory requirements of some very large computer algebra problems. Our aim is to achieve balanced memory use, which differentiates this work from other parallel systems whose focus is on gaining speedup. It is widely observed that failures in computer algebra systems are mostly due to memory overload: for several problems in computer algebra, some of the best available algorithms suffer from intermediate expression swell where the result is of reasonable size, but the intermediate calculation encounters severe memory limitations. This observation motivates our memory-centric approach to parallelizing computer algebra algorithms. The memory balancing is based on a randomized hashing algorithm for dynamic distribution of data. Dynamic distribution means that the intermediate data is allocated storage space at the time that it is created and therefore the system can avoid overloading some processing elements. Large scale computer algebra problems with peak memory demands of more than 10 gigabytes are considered. Distributed memory can scale to satisfy these requirements. For example, the Hitachi SR2201 which is the target architecture in this research provides up to 56 gigabytes of memory. The system has fine granularity : tasks sizes are small and data is partitioned in small blocks. The fine granularity provides flexibility in controlling memory balance but incurs higher communication costs. The communication overhead is reduced by an intelligent scheduler which performs asynchronous overlap of communication and computation. The implementation provides a polynomial algebra system with operations on multivariate polynomials and matrices with polynomial entries. Within this framework it is possible to find computations with large memory demands, for example, solving large sparse systems of linear equations and Gröbner base computations. The parallel algorithms that have been implemented are based on the standard algorithms for polynomial algebra. This demonstrates that careful attention to memory management aids solution of very large problems even without the benefit of advanced algorithms. The parallel implementation can be used to solve larger problems than have previously been possible.