Re: QR algorithm with explicit shifts
- From: spellucci@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx (Peter Spellucci)
- Date: Tue, 1 Nov 2005 10:03:21 +0000 (UTC)
In article <0Hu8f.5510$sA4.1069@xxxxxxxxxxxxxxxxxxxx>,
"BemusedbyQM" <groover892002@xxxxxxxxxxx> writes:
>hi,
>
>i am using the QR algorithm with explicit shifts to find the eigenvalues of
>a real or complex matrix.
>
>the way i have it set up at present is that it selects a shift (a wilkinson
>shift taken from the eigenvalues of the bottom right hand corner 2 x 2
>submatrix) , and then carries out iterations of the QR algorithm until one
>of the eigenvalues converges. it then stops, deflates the matrix, selects
>another shift and carries on with the QR algorithm until the next eigenvalue
>converges.
>
>my question is, would it be possible to select a new shift after every
>iteration , rather than waiting for convergence? if so would there be any
>advantage in doing this, with regards to how quickly overall convergence is
>acheived etc?
>
>
>thanks
>
>
normally the QR iteration is performed with a new (Wilkinson)shift at every
single iteration step.
only this technique leads to the supercubic convergence for the hermitian
tridiagonal case.
your approach makes sense in the nonhermitian case and for the first eigenvalue
because in the beginning there is little to say about the value of the
Wilkinsonshift. and do not forget the exceptional random shift in case of
no obvious convergence after a number of steps (30 in the old days).
hth
peter
.
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