Fast randomized least-squares solvers can be just as accurate and stable as classical direct solvers

One of the greatest success stories of randomized algorithms for linear algebra has been the development of fast, randomized algorithms for highly overdetermined linear least-squares problems. However, none of the existing algorithms is backward stable, preventing them from being deployed as drop-in replacements for existing QR-based solvers. This paper introduces FOSSILS, a fast, provably backward stable randomized least-squares solver. FOSSILS combines iterative refinement with a preconditioned iterative method applied to the normal equations, and it converges at the same rate as existing randomized least-squares solvers. This work offers the promise of incorporating randomized least-squares solvers into existing software libraries while maintaining the same level of accuracy and stability as classical solvers.