Abstract
This paper considers elimination methods to solve dense linear systems, in particular a variant of Gaussian elimination due to Huard [13]. This variant reduces the system to an equivalent diagonal system just like Gauss-Jordan elimination, but does not require more floating-point operations than Gaussian elimination. To preserve stability, a pivoting strategy using column interchanges, proposed by Hoffmann [10], is incorporated in the original algorithm. An error analysis is given showing that Huard’s elimination method is as stable as Gauss-Jordan elimination with the appropriate pivoting strategy. This result is proven in a similar way as the proof of stability for Gauss-Jordan elimination given in [4]. Numerical experiments are reported which verify the theoretical error analysis of the Gauss-Huard algorithm.
Zusammenfassung
Wir betrachten Eliminationsverfahren zur Lösung linearer Gleichungssysteme mit voll besetzter Koeffizientenmatriz und zwar besonders eine von Huard eingeführte Variante des Gauss’schen Algorithmus [13]. Dabei wird das Gleichungssystem auf Diagonalform reduziert wie in der Gauss-Jordan Variante des Eliminationsverfahrens, aber es werden nicht mehr Operationen benötigt als im klassischen Algorithmus von Gauss. Um die Stabilität zu garantieren, wird eine von Hoffmann entwickelte Pivotstrategie verwendet [10]. Eine Fehleranalyse zeigt, daß der Gauss-Huard Algorithmus kombiniert mit dieser Pivotstrategie genau so stabil ist wie der Gauss-Jordan Algorithmus. Der Beweis ist analog zum Beweis für die Stabilität des Gauss-Jordan Algorithmus in [4]. Numerische Resultate bestätigen die theoretisch gefundene Fehleranalyse.
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Dekker, T.J., Hoffmann, W. & Potma, K. Stability of the Gauss-Huard algorithm with partial pivoting. Computing 58, 225–244 (1997). https://doi.org/10.1007/BF02684391
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DOI: https://doi.org/10.1007/BF02684391