Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach

TL;DR

This paper introduces a mixed precision approach to enhance the accuracy and scalability of MRRR-based eigensolvers for symmetric tridiagonal eigenproblems, achieving better performance and precision.

Contribution

The paper demonstrates that mixed precision techniques significantly improve the accuracy of MRRR eigensolvers without sacrificing performance.

Findings

Mixed precision improves eigensolver accuracy.

Enhanced eigensolvers are faster and more scalable.

Achieves accuracy comparable or superior to existing methods.

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) is among the fastest methods. Although fast, the solvers based on MRRR do not deliver the same accuracy as competing methods like Divide & Conquer or the QR algorithm. In this paper, we demonstrate that the use of mixed precisions leads to improved accuracy of MRRR-based eigensolvers with limited or no performance penalty. As a result, we obtain eigensolvers that are not only equally or more accurate than the best available methods, but also -in most circumstances- faster and more scalable than the competition.