The algorithm of multiple relatively robust representations for multi-core processors

Abstract

The algorithm of Multiple Relatively Robust Representations (MRRR or MR 3) computes k eigenvalues and eigenvectors of a symmetric tridiagonal matrix in O(nk) arithmetic operations. Large problems can be effectively tackled with existing distributed-memory parallel implementations of MRRR; small and medium size problems can instead make use of LAPACK's routine xSTEMR. However, xSTEMR is optimized for single-core CPUs, and does not take advantage of today's multi-core and future many-core architectures. In this paper we discuss some of the issues and trade-offs arising in the design of MR 3-SMP, an algorithm for multi-core CPUs and SMP systems. Experiments on application matrices indicate that MR 3-SMP is both faster and obtains better speedups than all the tridiagonal eigensolvers included in LAPACK and Intel's Math Kernel Library (MKL). © 2012 Springer-Verlag.