A nested Krylov subspace method to compute the sign function of large complex matrices

TL;DR

This paper introduces a nested Krylov subspace method with preconditioning to efficiently compute the sign function of large complex matrices, significantly improving computational speed in lattice QCD simulations.

Contribution

It proposes a novel nested Krylov approach with preconditioning that accelerates sign function computations for large matrices, outperforming existing rational approximation methods.

Findings

Achieves high accuracy with smaller subspaces

Demonstrates substantial efficiency gains on large lattice configurations

Outperforms traditional rational approximation methods in speed

Abstract

We present an acceleration of the well-established Krylov-Ritz methods to compute the sign function of large complex matrices, as needed in lattice QCD simulations involving the overlap Dirac operator at both zero and nonzero baryon density. Krylov-Ritz methods approximate the sign function using a projection on a Krylov subspace. To achieve a high accuracy this subspace must be taken quite large, which makes the method too costly. The new idea is to make a further projection on an even smaller, nested Krylov subspace. If additionally an intermediate preconditioning step is applied, this projection can be performed without affecting the accuracy of the approximation, and a substantial gain in efficiency is achieved for both Hermitian and non-Hermitian matrices. The numerical efficiency of the method is demonstrated on lattice configurations of sizes ranging from 4^4 to 10^4, and the new results are compared with those obtained with rational approximation methods.