Carson, Erin and Higham, Nicholas J. (2017) A New Analysis of Iterative Refinement and its Application to Accurate Solution of IllConditioned Sparse Linear Systems. SIAM Journal on Scientific Computing, 39 (6). A2834A2856. ISSN 10957197
This is the latest version of this item.
Text
17m1122918.pdf  Accepted Version Available under License Creative Commons Attribution. Download (532kB) 
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:  Article 

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:  11 Dec 2017 22:48 
Last Modified:  11 Dec 2017 22:48 
URI:  http://eprints.maths.manchester.ac.uk/id/eprint/2604 
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)
 A New Analysis of Iterative Refinement and its Application to Accurate Solution of IllConditioned Sparse Linear Systems. (deposited 11 Dec 2017 22:48) [Currently Displayed]

A New Analysis of Iterative Refinement and its Application to
Accurate Solution of IllConditioned Sparse Linear Systems. (deposited 26 Jul 2017)
Actions (login required)
View Item 