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Purging $\theta $.

If $\theta $ is ``unwanted" then we may wish to remove $\theta $ from the spectrum of the projected matrix $H$. However, the implicit restart strategy using exact shifts will sometimes fail to purge a converged unwanted Ritz value [294].

We shall use (7.21) to purge an unwanted but converged Ritz value. In this case, a left eigenvector $y$ is needed with

\begin{displaymath}
y^*h = \theta y^*, \ \ \Vert y\Vert = 1.
\end{displaymath}

Now, when we apply $AVQ = VQ (Q^*H Q) + f e_k^* Q $, we obtain

\begin{displaymath}
A [v_1 , V_2] = [v_1 , V_2] = \left[
\ba{cc}
\theta & 0 \\
h & H_2
\ea
\right]
+ f (\eta, \tau e_{k-1}^*),
\end{displaymath}

where $H_2$ is upper Hessenberg. Here, $\eta = e_k^*y$ as before, but there is no requirement that $\eta$ be small. The desired purging amounts to simply discarding the first column on both sides of this equation. We are then left with

\begin{displaymath}
A V_2 = V_2 H_2 + f\tau e_{k-1}^*.
\end{displaymath}

No error other than an acceptable level of roundoff will be introduced through this purging process. Moreover, there is no requirement that $y$ be an accurate left eigenvector for $H$. It is only necessary that the residual $y^* H - \theta y^* $ be small.


next up previous contents index
Next: Stability of . Up: Orthogonal Deflating Transformation Previous: Locking .   Contents   Index
Susan Blackford 2000-11-20