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

Carson, Erin and Higham, Nicholas J. (2017) A New Analysis of Iterative Refinement and its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems. [MIMS Preprint]

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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-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 $3$ steps in every case.

Item Type: MIMS Preprint
Uncontrolled Keywords: ill-conditioned linear system, iterative refinement, multiple precision, mixed precision, rounding error analysis, backward error, forward error, GMRES, preconditioning
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 26 Jul 2017
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2563

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