these from the large number of subsequent right hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses.
author:
- 'Andreas Stathopoulos [^1]'
- 'Konstantinos Orginos [^2]'
title: 'Computing and deflating eigenvalues while solving multiple right hand side linear systems with an application to Quantum Chromodynamics [^3] '
---

**Keywords:** Hermitian linear systems, multiple right hand sides, eigenvalues, deflation, Lanczos, Conjugate Gradient

Introduction
============

The numerical solution of linear systems of equations of large, sparse matrices is central to many scientific and engineering applications. One of the most computationally demanding applications is lattice Quantum Chromodynamics (QCD) because not only does it involve very large matrix sizes but also requires the solution of several linear systems with the same matrix but different right hand sides. Direct methods, although attractive for multiple right hand sides, cannot be used because of the size of the matrix. Iterative methods provide the only means for solving these problems.

QCD is the theory of the fundamental force 