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The shift $ \kappa $

The choice of the shift is very important to achieve a fast convergence about one of the eigenvalues. The shifts we consider here are discussed in [91]. More references towards analysis of different shifts can also be found there. Suppose we have the following semiseparable matrix represented with the Givens-vector representation:

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$\displaystyle \left( \begin{array}{ccccc} c_1 d_1 ...
...-1}s_{n-2}\cdots s_1 d_1 & \hdots & & s_{n-1}d_{n-1} & d_n \end{array} \right).$    

One can choose $ d_n$ as a shift, the so-called Rayleigh shift, or one can consider as a shift the eigenvalue of

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$\displaystyle \left( \begin{array}{cc} c_{n-1}d_{n-1} & s_{n-1} d_{n-1} \\  s_{n-1}d_{n-1} & d_n \end{array} \right),$    

that is closest to $ d_n$, the Wilkinson shift [190]. Using this shift in the tridiagonal case will give cubic convergence. The numerical results provided in Chapter 9 will experimentally prove the same rate of convergence for the symmetric semiseparable case.


next up previous contents index
Next: An implicit -step on Up: An implicit -algorithm for Previous: Unreduced symmetric semiseparable matrix   Contents   Index
Raf Vandebril 2004-05-03