A new BLAS-3 based parallel algorithm for computing the eigenvectors of real symmetric matrices

Authors:
Yusaku Yamamoto;Mitsuyoshi Igai;Ken Naono
Affiliations:
Department of Computational Science and Engineering, Nagoya University, Japan;Hitachi ULSI Systems Corp. and Central Research Laboratory, Hitachi Ltd.;Hitachi ULSI Systems Corp. and Central Research Laboratory, Hitachi Ltd.
Venue:
High performance scientific and engineering computing
Year:
2004

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Abstract

We developed a new algorithm for computing the eigenvectors of a real symmetric matrix on shared-memory parallel computers. Instead of using the modified Gram-Schmidt orthogonalization, which is the bottleneck in parallelizing the conventional inverse iteration algorithm, we choose to hold the basis of orthogonal complementary subspace of the calculated eigenvectors explicitly, and successively modify it by the Householder transformations. This enables us to use the BLAS-2 routines instead of the BLAS-1 routines and reduce the number of interprocessor synchronization from O(N2) to O(N). The performance of the algorithm is further enhanced with the blocking technique, which allows the BLAS-2 routines to be replaced with the BLAS-3 routines. We evaluated our algorithm on 1 node of the SR8000 (a shared-memory parallel computer with 8 processors) and obtained performance 3.1 times higher than that of the conventional method when computing all the eigenvectors of a matrix of order 1000.