Rational QR Transformation with Newton Shift for Symmetric Tridiagonal Matrices
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 corrections in combination with QR steps. If an initial shift has rendered the matrix positive or negative definite, then this property is preserved throughout the iteration. Thus, the QR step may be achieved by two successive Cholesky LR steps or equivalently, since the matrix is tridiagonal, by two QD steps which are numerically stable  and avoid square roots. The rational QR step used here needs slightly fewer additions than the Ortega-Kaiser step .
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