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• W2599973081 abstract "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." @default.
• W2599973081 created "2017-04-07" @default.
• W2599973081 creator A5069339254 @default.
• W2599973081 creator A5072265294 @default.
• W2599973081 date "2017-01-01" @default.
• W2599973081 modified "2023-10-17" @default.
• W2599973081 title "A New Analysis of Iterative Refinement and Its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems" @default.
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• W2599973081 doi "https://doi.org/10.1137/17m1122918" @default.
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