Fast Blocking of Householder Reflectors on Graphics Processors

Abstract

We revisit an alternative representation to the compact WY transform for the accumulation (blocking) of Householder reflectors that exhibits the same numerical stability and is composed of efficient computational kernels from Level-3 Basic Linear Algebra Subprograms (BLAS) in contrast with the Level-2 BLAS that are utilized for the construction of the conventional compact WY representation. For the orthogonal reduction to condensed forms on multicore platforms equipped with a fast graphics processing unit (GPU), (or when there is a notable gap in performance between the multicore processors and the graphics accelerator,) our approach removes the assembly of the accumulation from the critical path of the algorithm. This comes as a consequence of accelerating this operation via the use of Level-3 BLAS, moving this computation to the GPU, and allowing the use of larger algorithmic block sizes. Our experiments with the alternative orthogonal representation show considerable speed-ups, which can be in the range 20-40% on recent GPUs when compared with the codes in MAGMA.