Two-sided orthogonal reductions to condensed forms on asymmetric multicore processors

Abstract

We investigate how to leverage the heterogeneous resources of an Asymmetric Multicore Processor (AMP) in order to deliver high performance in the reduction to condensed forms for the solution of dense eigenvalue and singular-value problems. The routines that realize this type of two-sided orthogonal reductions (TSOR) in LAPACK are especially challenging, since a significant fraction of their floating-point operations are cast in terms of memory-bound kernels while the remaining part corresponds to efficient compute-bound kernels. To deal with this scenario: (1) we leverage implementations of memory-bound and compute-bound kernels specifically tuned for AMPs; (2) we select the algorithmic block size for the TSOR routines via a practical model; and (3) we adjust the type and number of cores to use at each step of the reduction. Our experiments validate the model and assess the performance of our asymmetry-aware TSOR routines, using an ARMv7 big.LITTLE AMP, for three key operations: the reduction to tridiagonal form for symmetric eigenvalue problems, the reduction to Hessenberg form for non-symmetric eigenvalue problems, and the reduction to bidiagonal form for singular-value problems.