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)


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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© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • C. Reinsch
  • F. L. Bauer

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