Extending the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides

TL;DR

This paper extends the eigCG algorithm to non-symmetric linear systems with multiple right-hand sides, demonstrating promising results in Lattice QCD problems and addressing critical slowing down.

Contribution

It introduces a novel extension of eigCG to non-symmetric systems using the two-sided Lanczos algorithm, enabling efficient eigenvalue computation and system solving.

Findings

Significant speed-ups in Lattice QCD simulations

Effective removal of critical slowing down

Promising results for gamma5-Hermitian systems

Abstract

For Hermitian positive definite linear systems and eigenvalue problems, the eigCG algorithm is a memory efficient algorithm that solves the linear system and simultaneously computes some of its eigenvalues. The algorithm is based on the Conjugate-Gradient (CG) algorithm, however, it uses only a window of the vectors generated by the CG algorithm to compute approximate eigenvalues. The number and accuracy of the eigenvectors can be increased by solving more right-hand sides. For Hermitian systems with multiple right-hand sides, the computed eigenvectors can be used to speed up the solution of subsequent systems. The algorithm was tested on Lattice QCD problems by solving the normal equations and was shown to give large speed up factors and to remove the critical slowing down as we approach light quark masses. Here, an extension to the non-symmetric case based on the two-sided Lanczos algorithm is given. The new algorithm is tested on Lattice QCD problems and is shown to give promising results. We also study the removal of the critical slowing down and compare results with those of the eigCG algorithm. We also discuss the case when the system is gamma5-Hermitian.