Numerical study of mixed precision GMRES(m) preconditioned by deflation

Abstract We present a numerical investigation on the mixed precision Generalized Minimal Residual (GMRES) method, preconditioned by deflation. GMRES is a widely used iterative solver for sparse, non-symmetric linear systems. The mixed precision version of restarted GMRES, or mixed precision GMRES(m), is known to achieve double precision accuracy while delivering substantial speedups by using single precision (32-bit) arithmetic for most computations and double precision (64-bit) for key corrective steps only. In this study, the mixed precision GMRES(m) is enhanced with a deflation-based preconditioner to accelerate its convergence. All preconditioning operations are carried out in single precision. We demonstrate the effectiveness of the mixed precision GMRES(m) preconditioned by deflation in terms of both convergence (iterations required) and computational performance. This is validated through numerical experiments on a multi-core environment using a variety of matrices. The results show that the preconditioner remains highly effective in reducing iterations even under reduced precision. Additionally, we compare the speedups achieved by mixed precision over double precision in the preconditioned GMRES(m) with those in the non-preconditioned version. For the majority of tested matrices, the speedups in the preconditioned and non-preconditioned cases are comparable.