# Linear systems solvers - recent developments and implications for lattice computations

## TL;DR

This paper reviews recent advances in Krylov subspace methods for solving non-Hermitian linear systems, emphasizing their near-optimal performance for lattice gauge theory computations and highlighting preconditioning as a key area for future improvements.

## Contribution

It analyzes the effectiveness of mature Krylov methods like QMR, BiCGStab, and GMRES for Wilson fermion matrices, stressing the importance of preconditioning.

## Key findings

- Krylov methods are near-optimal for Wilson fermion matrices
- Preconditioning is crucial for further improvements
- Mature methods like QMR, BiCGStab, GMRES are effective

## Abstract

We review the numerical analysis' understanding of Krylov subspace methods for solving (non-hermitian) systems of equations and discuss its implications for lattice gauge theory computations using the example of the Wilson fermion matrix. Our thesis is that mature methods like QMR, BiCGStab or restarted GMRES are close to optimal for the Wilson fermion matrix. Consequently, preconditioning appears to be the crucial issue for further improvements.