Carson, Erin and Higham, Nicholas J. (2017) A New Analysis of Iterative Refinement and its Application to Accurate Solution of IllConditioned Sparse Linear Systems. [MIMS Preprint]
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Abstract
Iterative refinement is a longstanding 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 GMRESbased iterative refinement method (GMRESIR) that makes use of the computed LU factors as preconditioners. GMRESIR 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 GMRESIR 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 matricesboth random and from the University of Florida Sparse Matrix CollectionGMRESIR yields a normwise relative error of order $u$ in at most $3$ steps in every case.
Item Type:  MIMS Preprint 

Uncontrolled Keywords:  illconditioned 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 
Available Versions of this Item

A New Analysis of Iterative Refinement and its Application to
Accurate Solution of IllConditioned Sparse Linear Systems. (deposited 28 Mar 2017)
 A New Analysis of Iterative Refinement and its Application to Accurate Solution of IllConditioned Sparse Linear Systems. (deposited 26 Jul 2017) [Currently Displayed]
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