A New Analysis of Iterative Refinement and Its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems

Iterative refinement is a long-standing technique for improving the accuracy of a computed solution to a nonsingular linear system $Ax = b$ obtained via LU factorization. It makes use of residuals computed in extra precision, typically at twice the working precision, and existing results guarantee convergence if the matrix A has condition number safely less than the reciprocal of the unit roundoff, u. We identify a mechanism that allows iterative refinement to produce solutions with normwise relative error of order u to systems with condition numbers of order u⁻¹ or larger, provided that the update equation is solved with a relative error sufficiently less than 1. A new rounding error analysis is given, and its implications are analyzed. Building on the analysis, we develop a GMRES (generalized minimal residual)-based iterative refinement method (GMRES-IR) that makes use of the computed LU factors as preconditioners. GMRES-IR exploits the fact that even if A is extremely ill conditioned the LU factors contain enough information that preconditioning can greatly reduce the condition number of A. Our rounding error analysis and numerical experiments show that GMRES-IR can succeed where standard refinement fails, and that it can provide accurate solutions to systems with condition numbers of order u⁻¹ and greater. Indeed, in our experiments with such matrices---both random and from the University of Florida Sparse Matrix Collection---GMRES-IR yields a normwise relative error of order u in at most three steps in every case.