Exploiting the parallel divide-and-conquer method to solve the symmetric tridiagonal eigenproblem

In this paper we present a new divide-and-conquer parallel algorithm to compute the eigenvalues of symmetric tridiagonal matrices. This algorithm combines the use of rank-one modifications in the division phase and the application of the Laguerre iteration in the updating phase. Our method is compared with one based on the same scheme but using rank-two modifications. A thorough experimental analysis in the Cray T3D parallel computer has been carried out. Special emphasis has been put on analysing the influence of the deflation phenomena on the computational cost of this kind of algorithm. Experimental results show that an adequate exploitation of the inherent parallelism in the divide-and-conquer scheme produces very efficient parallel algorithms. The obtained speedups clearly improve the best sequential algorithm, including the standard implementation of QR iteration in LAPACK.