High-Performance Solvers for Dense Hermitian Eigenproblems

TL;DR

This paper presents EleMRRR, a new collection of high-performance, scalable solvers for large dense Hermitian eigenproblems, including a fast tridiagonal solver, outperforming existing ScaLAPACK solutions on supercomputers.

Contribution

Introduction of EleMRRR, a scalable and efficient solver suite for dense Hermitian eigenproblems, with a novel tridiagonal solver based on the PMRRR algorithm, integrated into the Elemental library.

Findings

EleMRRR outperforms ScaLAPACK solvers on supercomputers.

The tridiagonal solver PMRRR is fast and scalable.

Guidelines for assembling the fastest solver within ScaLAPACK.

Abstract

We introduce a new collection of solvers - subsequently called EleMRRR - for large-scale dense Hermitian eigenproblems. EleMRRR solves various types of problems: generalized, standard, and tridiagonal eigenproblems. Among these, the last is of particular importance as it is a solver on its own right, as well as the computational kernel for the first two; we present a fast and scalable tridiagonal solver based on the Algorithm of Multiple Relatively Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers, PMRRR is part of the freely available Elemental library, and is designed to fully support both message-passing (MPI) and multithreading parallelism (SMP). As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's solvers on two supercomputers. Such a study, performed with up to 8,192 cores, provides precise guidelines to assemble the fastest solver within the ScaLAPACK framework; it also indicates that EleMRRR outperforms even the fastest solvers built from ScaLAPACK's components.