Multitasking the Davidson Algorithm for the Large, Sparse Eigenvalue Problem

Abstract

The Davidson algorithm, which was developed for han dling the eigenvalue problem for large and sparse ma trices arising in quantum chemistry, was modified for use in atomic structure calculations. To date these cal culations have used traditional eigenvalue methods, which limit the range of feasible calculations because of their excessive memory requirements and unsatisfactory performance attributed to time-consuming and costly processing of zero valued elements. The replacement of a traditional matrix eigenvalue method by the Davidson algorithm reduced these limitations. Significant speedup was found, which varied with the size of the underlying problem and its sparsity. Furthermore, the range of ma trix sizes that can be manipulated efficiently was ex panded by more than one order of magnitude. On the CRAY X-MP the code was vectorized and the importance of gather/scatter analyzed. A parallelized version of the algorithm obtained an additional 35% reduction in exe cution time. Speedup due to vectorization and concur rency was also measured on the Alliant FX/8.