Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics

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Abstract
We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with Conjugate Gradient (CG). The algorithm capitalizes on the vectors already available from CG, building a small window of vectors that approximates the eigenvectors. While this window is restarted in a locally optimal way, the CG is not. Our algorithm converges almost identically to unrestarted Lanczos, yet without the need to store all Lanczos vectors. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. When solving systems with multiple right hand sides, eigenvectors identified in earlier linear systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice QCD applications, where hundreds of right hand sides may be needed. Specifically, our deflation technique can remove 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.

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