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On a modification of the QZ algorithm with fast givens rotations

Eine Modifikation des QZ-Algorithmus mit schnellen Givens-Rotationen

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Abstract

It is explained, how Householder reflections can be replaced by fast Givens rotations in the first and second step of the QZ algorithm. A count of the required operations for the two steps shows, that the modified first step does need less operations, but that the modified second step needs only insignificantly less operations in comparison with the original QZ algorithm. On the base of this fact a modification of the second step is proposed. The resulting new algorithm is called FQZ algorithm, which has the same numerical properties as the QZ algorithm, if a correct scaling is applied.

Zusammenfassung

Es wird dargestellt, auf welche Weise im ersten und zweiten Schritt des QZ-Algorithmus Householder-Transformationen durch schnelle Givens-Rotationen ersetzt werden können. Die Bestimmung des Rechenaufwandes für diese beiden Schritte zeigt, daß sich im ersten Schritt gegenüber dem ursprünglichen Algorithmus Operationen einsparen lassen, im zweiten Schritt jedoch nur eine unwesentliche Einsparung möglich ist. Auf Grund dieser Tatsache wird der zweite Schritt modifiziert und der daraus enstehende neue Algorithmus als FQZ-Algorithmus bezeichnet. Falls korrekt skaliert wird, entsprechen die numerischen Eigenschaften des FQZ-Algorithmus denjenigen des originalen QZ-Algorithmus.

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Authors and Affiliations

  1. Institut für angewandte Mathematik, Universität Zürich, Rämistrasse 74, CH-8001, Zürich, Switzerland

    H. P. Märchy

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  1. H. P. Märchy
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Märchy, H.P. On a modification of the QZ algorithm with fast givens rotations. Computing 38, 247–259 (1987). https://doi.org/10.1007/BF02240099

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