Blocked algorithms for the reduction to Hessenberg-triangular form revisited

We present two variants of Moler and Stewart's algorithm for reducing a matrix pair to Hessenberg-triangular (HT) form with increased data locality in the access to the matrices. In one of these variants, a careful reorganization and accumulation of Givens rotations enables the use of efficient level 3 BLAS. Experimental results on four different architectures, representative of current high performance processors, compare the performances of the new variants with those of the implementation of Moler and Stewart's algorithm in subroutine DGGHRD from LAPACK, Dackland and Kågström's two-stage algorithm for the HT form, and a modified version of the latter which requires considerably less flop

Accelerating Computation of Eigenvectors in the Dense Nonsymmetric Eigenvalue Problem

Abstract. In the dense nonsymmetric eigenvalue problem, work has focused on the Hessenberg reduction and QR iteration, using efficient al-gorithms and fast, Level 3 BLAS. Comparatively, computation of eigen-vectors performs poorly, limited to slow, Level 2 BLAS performance with little speedup on multi-core systems. It has thus become a dominant cost in the solution of the eigenvalue problem. To address this, we present im-provements for the eigenvector computation to use Level 3 BLAS and parallelize the triangular solves, achieving good parallel scaling and ac-celerating the overall eigenvalue problem more than three-fold.