Numerical Methods for the QCD Overlap Operator: I. Sign-Function and Error Bounds

TL;DR

This paper evaluates various numerical methods for computing the sign-function of matrices in lattice QCD, providing error bounds and comparisons to improve computational efficiency and accuracy.

Contribution

It introduces and compares Lanczos and partial fraction expansion methods with error bounds for the overlap operator in lattice QCD.

Findings

Lanczos-based methods are effective for sign-function computation.

Error bounds enable guaranteed accuracy in iterative methods.

Comparative analysis guides method selection for lattice QCD simulations.

Abstract

The numerical and computational aspects of the overlap formalism in lattice quantum chromodynamics are extremely demanding due to a matrix-vector product that involves the sign function of the hermitian Wilson matrix. In this paper we investigate several methods to compute the product of the matrix sign-function with a vector, in particular Lanczos based methods and partial fraction expansion methods. Our goal is two-fold: we give realistic comparisons between known methods together with novel approaches and we present error bounds which allow to guarantee a given accuracy when terminating the Lanczos method and the multishift-CG solver, applied within the partial fraction expansion methods.