MRRR-based Eigensolvers for Multi-core Processors and Supercomputers

TL;DR

This paper discusses the MRRR algorithm for solving real symmetric tridiagonal eigenproblems, highlighting its efficiency and accuracy, especially for computing multiple eigenpairs on modern multi-core and supercomputing architectures.

Contribution

It provides an in-depth analysis of MRRR's performance and accuracy, emphasizing its advantages over other methods in large-scale eigenproblem computations.

Findings

MRRR requires O(kn) operations for k eigenpairs, making it faster than other methods.

MRRR maintains high accuracy in eigenvalue and eigenvector computations.

The paper evaluates MRRR's scalability and effectiveness on multi-core processors and supercomputers.

Abstract

The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to tridiagonal form. For its solution, the algorithm of Multiple Relatively Robust Representations (MRRR or MR3 in short) - introduced in the late 1990s - is among the fastest methods. To compute k eigenpairs of a real n-by-n tridiagonal T, MRRR only requires O(kn) arithmetic operations; in contrast, all the other practical methods require O(k^2 n) or O(n^3) operations in the worst case. This thesis centers around the performance and accuracy of MRRR.