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IMA Journal of Numerical Analysis 1997 17(4):495-509; doi:10.1093/imanum/17.4.495
© 1997 by Institute of Mathematics and its Applications
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Iterative refinement for linear systems and LAPACK

NICHOLAS J. HIGHAM{dagger}

Department of Mathematics, University of Manchester Manchester M13 9PL, UK

{dagger} E-mail: na.nhigham{at}na-net.ornl.gov

The technique of iterative refinement for improving the computed solution to a linear system was used on desk calculators and computers in the 1940s and has remained popular. In the 1990s iterative refinement is well supported in software libraries, notably in LAPACK. Although the behaviour of iterative refinement in floating point arithmetic is reasonably well understood, the existing theory is not sufficient to justify the use of fixed precision iterative refinement in all the LAPACK routines in which it is implemented. We present analysis that provides the theoretical support needed for LAPACK. The analysis covers both mixed and fixed precision iterative refinement with an arbitrary number of iterations, makes only a general assumption on the underlying solver, and is relatively short. We identify some remaining open problems.


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