On Aggressive Early Deflation in Parallel Variants of the QR Algorithm

Abstract

The QR algorithm computes the Schur form of a matrix and is by far the most popular approach for solving dense nonsymmetric eigenvalue problems. Multishift and aggressive early deflation (AED) techniques have led to significantly more efficient sequential implementations of the QR algorithm during the last decade. More recently, these techniques have been incorporated in a novel parallel QR algorithm on hybrid distributed memory HPC systems. While leading to significant performance improvements, it has turned out that AED may become a computational bottleneck as the number of processors increases. In this paper, we discuss a two-level approach for performing AED in a parallel environment, where the lower level consists of a novel combination of AED with the pipelined QR algorithm implemented in the ScaLAPACK routine PDLAHQR. Numerical experiments demonstrate that this new implementation further improves the performance of the parallel QR algorithm.