Accelerating the reduction to upper Hessenberg form through hybrid GPU-based computing

We present a Hessenberg reduction (HR) algorithm for hybrid multicore + GPU systems that gets more than 16× performance improvement over the current LAPACK algorithm running just on current multicores (in double precision arithmetic). This enormous acceleration is due to proper matching of algorithmic requirements to architectural strengths of the hybrid components. The reduction itself is an important linear algebra problem, especially with its relevance to eigenvalue problems. The results described in this paper are significant because Hessenberg reduction has not yet been accelerated on multicore architectures, and it plays a significant role in solving nonsymmetric eigenvalue problems. The approach can be applied to the symmetric problem and in general, to two-sided matrix transformations. The work further motivates and highlights the strengths of hybrid computing: to harness the strengths of the components of a hybrid architecture to get significant computational acceleration which otherwise may have been impossible.