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Rational QR Transformation with Newton Shift for Symmetric Tridiagonal Matrices

  • C. Reinsch
  • F. L. Bauer
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)

Abstract

If some of the smallest or some of the largest eigenvalues of a symmetric (tridiagonal) matrix are wanted, it suggests itself to use monotonic Newton corput rections in combination with Q R steps. If an initial shift has rendered the matrix positive or negative definite, then this property is preserved throughout the iteration. Thus, the Q R step may be achieved by two successive Cholesky L R steps or equivalently, since the matrix is tridiagonal, by two Q D steps which are numerically stable [4] and avoid square roots. The rational Q R step used here needs slightly fewer additions than the Ortega-Kaiser step [3].

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References

  1. 1.
    Barth, W., R. S. Martin, and J. H. Wilkinson: Handbook series linear algebra. Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection. Numer. Math. 9, 386–393 (1967). Cf. II/5.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kahan, W.: Accurate eigenvalues of a symmetric tri-diagonal matrix. Technical Report No. CS 41, Computer Science Department, Stanford University 1966.Google Scholar
  3. 3.
    Ortega, J. M., and H. F. Kaiser. The LLT and QR methods for symmetric tridiagonal matrices. Comput. J. 6, 99–101 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Rutishauser, H.: Stabile Sonderfälle des Quotienten-Differenzen-Algorithmus. Numer. Math. 5, 95–112 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Rutishauser, H. and H. R. Schwarz. The LR transformation method for symmetric matrices. Numer. Math. 5, 273–289 (1963).MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Wilkinson, J. H.: Rounding errors in algebraic processes. Notes on applied science No. 32. London: Her Majesty’s Stationary Office 1963. German edition: Rundungsfehler. Berlin-Göttingen-Heidelberg: Springer 1969.Google Scholar
  7. 7.
    Wilkinson The algebraic eigenvalue problem. Oxford: Clarendon Press 1965-zbMATHGoogle Scholar
  8. 8.
    Bauer, F. L.: QD-method with Newton shift. Techn. Rep. 56. Computer Science Department, Stanford University 1967.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • C. Reinsch
  • F. L. Bauer

There are no affiliations available

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