An Efficient Parallel Algorithm for Extreme Eigenvalues of Sparse Nonsymmetric Matrices

Main memory accesses for shared-memory systems or global communications (synchronizations) in message passing systems decrease the computation speed. In this paper, the standard Arnoldi algorithm for approxi mating a small number of eigenvalues, with largest (or smallest) real parts for nonsymmetric large sparse ma trices, is restructured so that only one synchronization point is required; that is, one global communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory ma chine per each iteration is required. We also introduce an s-step Arnoldi method for finding a few eigenvalues of nonsymmetric large sparse matrices. This method generates reduction matrices that are similar to those generated by the standard method. One iteration of the s-step Arnoldi algorithm corresponds to s itera tions of the standard Arnoldi algorithm. The s-step method has improved data locality, minimized global communication, and superior parallel properties. These algorithms are implemented on a 64-node NCUBE/7 Hypercube and a CRAY-2, and performance results are presented.

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