Abstract
Many iterative methods for solving linear equationsAx=b aim for accurate approximations tox, and they do so by updating residuals iteratively. In finite precision arithmetic, these computed residuals may be inaccurate, that is, they may differ significantly from the (true) residuals that correspond to the computed approximations. In this paper we will propose variants on Neumaier's strategy, originally proposed for CGS, and explain its success. In particular, we will propose a more restrictive strategy for accumulating groups of updates for updating the residual and the approximation, and we will show that this may improve the accuracy significantly, while maintaining speed of convergence. This approach avoids restarts and allows for more reliable stopping criteria. We will discuss updating conditions and strategies that are efficient, lead to accurate residuals, and are easy to implement. For CGS and Bi-CG these strategies are particularly attractive, but they may also be used to improve Bi-CGSTAB, BiCGstab(l), as well as other methods.
Zusammenfassung
Viele iterative Methoden zur Lösung linearer Gleichungssysteme berechnen die Iterierten über aufdatierte Residuen. In endlicher Arithmetik können diese Residuen sehr ungenau sein, d.h., sie können sich erheblich von den tatsächlichen unterscheiden. In dieser Arbeit stellen wir Varianten der Neumaier Strategie vor, die ursprünglich für das CGS-Verfahren vorgeschlagen wurde, und erklären deren Erfolge. Insbesondere werden wir eine Variante vorschlagen, bei der mehrere Aufdatierungsschritte zusammengefaßt werden. Wir zeigen, daß sich die Genauigkeit der berechneten Residuen dadurch erheblich verbessern läßt, ohne daß die Konvergenzgeschwindigkeit beeinträchtigt wird. Dieser Ansatz vermeidet Neustarts und ermöglicht zuverlässigere Abbruchkriterien. Wir diskutieren Aufdatierungsbedingungen und Strategien, die effizient und leicht zu implementieren sind. Diese Strategien führen zu genaueren Residuen und sind insbesondere für CGS und Bi-CG-aber auch für Bi-CGSTAB, BiCGstab(l) und andere Verfahren-sehr attraktiv.
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Sleijpen, G.L.G., van der Vorst, H.A. Reliable updated residuals in hybrid Bi-CG methods. Computing 56, 141–163 (1996). https://doi.org/10.1007/BF02309342
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DOI: https://doi.org/10.1007/BF02309342