  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.
