On Implementation and Evaluation of Inverse Iteration Algorithm with Compact WY Orthogonalization

Abstract

A new inverse iteration algorithm that can be used to compute all the eigenvectors of a real symmetric tri-diagonal matrix on parallel computers is developed. The modified Gram-Schmidt orthogonalization is used in the classical inverse iteration. This algorithm is sequential and causes a bottleneck in parallel computing. In this paper, the use of the compact WY representation is proposed in the orthogonalization process of the inverse iteration with the Householder transformation. This change results in drastically reduced synchronization cost in parallel computing. The new algorithm is evaluated on both an 8-core and a 32-core parallel computer, and it is shown that the new algorithm is greatly faster than the classical inverse iteration algorithm in computing all the eigenvectors of matrices with several thousand dimensions.