A comparative study of numerical methods for the overlap Dirac operator--a status report

TL;DR

This paper compares numerical methods for computing the sign function in the overlap Dirac operator, highlighting an optimal partial fraction expansion and benchmarking its efficiency and accuracy.

Contribution

It introduces an optimal partial fraction expansion based on Zolotarev's theorem and benchmarks its performance against other methods.

Findings

The Zolotarev-based PFE is most efficient for approximating the sign function.

Removing converged systems improves computational efficiency.

A posteriori error bounds are provided for accuracy assessment.

Abstract

Improvements of various methods to compute the sign function of the hermitian Wilson-Dirac matrix within the overlap operator are presented. An optimal partial fraction expansion (PFE) based on a theorem of Zolotarev is given. Benchmarks show that this PFE together with removal of converged systems within a multi-shift CG appears to approximate the sign function times a vector most efficiently. A posteriori error bounds are given.