Bounded Parallelism in Computer Algebra

This thesis examines the performance improvements that can be made by exploiting parallel processing in symbolic mathematical computation. The study focuses on the use of high-level parallelism in the case where the number of processors is fixed and independent of the problem size, as in existing multiprocessors. Since seemingly small changes to the inputs can cause dramatic changes in the execution times of many algorithms in computer algebra, it is not generally useful to use static scheduling. We find it is possible, however, to exploit the high-level parallelism in many computer algebra problems using dynamic scheduling methods in which subproblems are treated homogeneously. Our investigation considers the reduction of execution time in both the case of ANDparallelism, where all of the subproblems must be completed, and the less well studied case of OR-parallelism, where completing any one of the subproblems is sufficient. We examine the use of AND and OR-parallelism in terms of the problem heap and collusive dynamic scheduling schemes which allow a homogeneous treatment of subtasks. A useful generalization is also investigated in which each of the subtasks may either succeed or fail and execution completes when the first success is obtained. We study certain classic problems in computer algebra within this framework. A collusive method for integer factorization is presented. This method attempts to find different factors in parallel, taking the first one that is discovered. Problem heap algorithms are given for the computation of multivariate polynomial GCDs and the computation of Gröbner bases. The GCD algorithm is based on the sparse modular GCD and performs the interpolations in parallel. The Gröbner basis algorithm exploits the independence of the reductions in basis completion to obtain a parallel method. In order to make evaluations in concrete terms, we have constructed a system for running computer algebra programs on a multiprocessor. The system is a version of Maple able to distribute processes over a local area network. The fact that the multiprocessor is a local area network need not be considered by the programmer.