Parallel Two-Stage Hessenberg Reduction using Tile Algorithms for Multicore Architectures

This paper describes a parallel Hessenberg reduction in the context of multicore architectures using tile algorithms. The Hessenberg reduction is very often used as a pre-processing step in solving dense linear algebra problems, such as the standard eigenvalue problem. Although expensive, orthogonal transformations are accepted techniques and commonly used for this reduction because they guarantee stability, as opposed to elementarily transformations similar to what is used in Gaussian elimination. The state of the art, high performance dense linear algebra software libraries, i.e., LAPACK and ScaLAPACK, reduce the matrix to Hessenberg form through a one-stage process by using block Householder approach with compact WY representation. However, the main drawback of the tile algorithms approach for the Hessenberg reduction is that the full reduction can not be obtained similarly in a one-stage standard process. A two-stage approach has to be considered. The first stage corresponds to a redesign of the block Hessenberg matrix reduction, introduced by Dongarra et. al [12], to benefit from tile algorithms in the multicore environment. The second stage further reduces the matrix bandwidth to achieve the required Hessenberg form using a parallel bulge chasing procedure. On the one hand, by exploiting the concepts of tile algorithms in the multicore environment, the block Hessenberg reduction (first stage) achieves 72% of the DGEMM peak on a 12000× 12000 matrix with 16 Intel Tigerton 2.4 GHz processors. On the other hand, the parallel bulge chasing procedure (second stage) is not appropriate for tile algorithms and therefore poorly performs on multicore architectures and slows down dramatically the overall algorithm.