Minimum residual methods for augmented systems |
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Authors: | B Fischer A Ramage D J Silvester A J Wathen |
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Institution: | (1) Institute of Mathematics, Medical University of Lübeck, Lübeck, Germany;(2) Department of Mathematics, University of Strathclyde, G1 1XH Glasgow, Scotland;(3) Department of Mathematics, UMIST, P. O. Box 88, M60 1QD Manchester, England;(4) Oxford University Computing Laboratory, OX1 3QD Oxford, England |
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Abstract: | For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability
and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems,
namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite
versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite
preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning)
and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean
norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations
for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where
the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this
case.
This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative
Research Grant CRG 960782 |
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Keywords: | 65F10 65F50 |
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