Stable Iterative Solvers for Ill-Conditioned Linear Systems and Least Squares

Iterative solvers for large-scale linear systems and least squares, such as Krylov subspace and iterative refinement methods, can diverge when the linear system is ill-conditioned, thus significantly reducing the applicability of these iterative methods in practice for high-performance computing solutions of such large-scale linear systems and least squares. To address this fundamental problem, we propose general algorithmic frameworks to modify broad classes of iterative solution methods which ensure that the algorithms are stable and do not diverge. We then apply our general frameworks to current implementations of iterative refinement and corresponding Krylov subspace iterative methods in SciPy, and demonstrate the efficacy of our stable iterative approach on linear-system and least-squares problems.