Multistage mixed precision iterative refinement

Low precision arithmetic, in particular half (16-bit) floating point is now available commercial hardware. Using lower can offer significant savings computation and communication costs with proportional energy. Motivated by this, there has been a renewed interest mixed iterative refinement schemes for solving linear systems A x = b , new variants of GMRES-based have developed. Each variant given combination precisions leads to different condition number-based constraints convergence the backward forward errors, each performance costs. The literature are, as an artifact analyses, often overly strict practice, thus could lead user select more expensive when less one would sufficed. In this work, we develop multistage solver which aims combine existing approaches balance accuracy improve usability. For user-specified initial precisions, algorithm begins least approach monitored via inexpensive computations quantities produced during iteration. If slow or divergence detected using stopping criteria, switches use expensive, but reliable variant. novel aspect our that, unlike implementations, first attempts “stronger” solvers solution update before resorting increasing precision(s). some scenarios, avoid need refactorize matrix higher precision. We perform extensive numerical experiments on variety random dense problems from real applications confirm benefits approach.