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Numerische Mathematik

, Volume 40, Issue 1, pp 47–56 | Cite as

Fast Givens rotations for orthogonal similarity transformations

  • Wolfgang Rath
Article

Summary

Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.

Subject Classifications

AMS (MOS): 65F30 CR: 5.14 

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Literature

  1. 1.
    Francis, J.C.F.: The QR-Transformation — A Unitary Analogue to the LR-Transformation Part I and II. Comput. J.4, 265–271 and 332–345 (1961/62)Google Scholar
  2. 2.
    Gentleman, W.M.: Least Squares Computations by Givens Transformations Without Square Roots. J. Inst. Math. Appl.12, 329–336 (1973)Google Scholar
  3. 3.
    Hammarling, S.: A Note on Modifications to the Givens Plane Rotations. J. Inst. Math. Appl.13, 215–218 (1974)Google Scholar
  4. 4.
    Moler, C.B., Stewart, G.W.: An Algorithm for Generalized Matrix Eigenvalue Problems. SIAM J. Numer. Anal.10, 241–256 (1973)Google Scholar
  5. 5.
    Rath, W.: Orthogonale Ähnlichkeitstransformationen mit schnellen Givensrotationen und ihre Anwendung für das Jacobiverfahren und den QR-Algorithmus. Diplomarbeit, Universität Bielefeld (West-Germany), May 1980Google Scholar
  6. 6.
    Stewart, G.W.: The Economical Storage of Plane Rotations. Numer. Math.25, 137–138 (1976)Google Scholar
  7. 7.
    Wilkinson, J.H., Reinsch, C.: Linear Algebra. Heidelberg, 1971Google Scholar
  8. 8.
    Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Oxford: University Press, 1965Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Wolfgang Rath
    • 1
  1. 1.AhlenGermany

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