On the implementation of a fully parallel algorithm for the symmetric eigenvalue problem

In this paper we present a parallel algorithm for the symmetric algebraic eigenvalue problem. The algorithm is based upon a divide and conquer scheme suggested by Cuppen for computing the eigensystem of a symmetric tridiagonal matrix. We extend this idea to obtain a parallel algorithm that retains a number of active parallel processes that is greater than or equal to the initial number throughout the course of the computation. Computation of the eigensystem of the tridiagonal matrix is reviewed. Also, brief analysis of the numerical properties and sensitivity to round off error is presented to indicate where numerical difficulties may occur. We show how to explicitly overlap the initial reduction to tridiagonal form with the parallel computation of the eigensystem of the tridiagonal matrix. The algorithm is therefore able to exploit parallelism at all levels of the computation and is well suited to a variety of architectures. Computational results have been presented in [4] for several machines. These results are very encouraging with respect to both accuracy and speedup. A surprising result is that the parallel algorithm for the tridiagonal case, even when run in serial mode, can be significantly faster than the previously best sequential algorithm on large problems, and is effective on moderate size problems when run in serial mode.