Krylov subspace methods and the sign function: multishifts and deflation in the non-Hermitian case

TL;DR

This paper develops efficient iterative methods for computing the matrix sign function of non-Hermitian matrices using rational approximations, multishifts, restarts, and deflation, demonstrated through numerical experiments on large lattice problems.

Contribution

It introduces a novel combination of multishift methods, restarts, and deflation for non-Hermitian matrices to improve computational efficiency.

Findings

Effective handling of large non-Hermitian matrices up to size 10^4.

Numerical experiments show improved performance for lattice QCD problems.

Restarts and deflation significantly reduce storage and computational costs.

Abstract

Rational approximations of the matrix sign function lead to multishift methods. For non-Hermitian matrices long recurrences can cause storage problems, which can be circumvented with restarts. Together with deflation we obtain efficient iterative methods, as we show in numerical experiments for the overlap Dirac operator at non-vanishing quark chemical potential for lattices up to size 10^4.