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The Lanczos algorithm is most commonly used in approximating a small number of extreme eigenvalues and eigenvectors for symmetric large sparse 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 Lanczos algorithm 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 communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory machine per each iteration is required.

We also introduce the s-step Lanczos method for finding a few eigenvalues of symmetric large sparse matrices in a similar way to the s-step Conjugate Gradient method [2], and we prove that the s-step method generates reduction …