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Locking $\theta $.

The first instance to discuss is the locking of a single converged Ritz value. Assume that

\begin{displaymath}
Ty = y \theta, \ \ \Vert y\Vert = 1,
\end{displaymath}

with $e_k^* y = \eta$, where $ \vert\eta \vert \le \epsilon_D \Vert T \Vert $. Here, it is understood that $\epsilon_M \le \epsilon_D < 1$ is a specified relative accuracy tolerance between $\epsilon_M$ and $1$.

If $\theta $ is ``wanted,'' it is desirable to lock $\theta $. However, in order to accomplish this, it will be necessary to arrange a transformation of the current Lanczos factorization to one with a small subdiagonal to isolate $\theta $. This may be accomplished by constructing a $k \times k$ orthogonal matrix $Q = Q(y)$ using Algorithm 4.9:

\begin{displaymath}
Q e_1 = y \ \ \mbox{and} \ \
e_k^* Q = ( \eta, \tau e_{k-1}^*),
\end{displaymath}

with $\eta^2 + \tau^2 = 1 $.

The end result of these transformations is Av_1 &=& v_1 + r , where v_1^* r = 0,
A V_2 &=& V_2 T_2 + r e_k-1^* , where $[v_1 , V_2] = V Q$.

This means that subsequent implicit restart takes place as if

\begin{displaymath}
A V_2 = V_2 T_2 + r\tau e_{k-1}^*
\end{displaymath}

with all the subsequent orthogonal transformations associated with implicit restart applied to $T_2$ and never disturbing the relation $ Av_1 = v_1 \theta + r \eta $. In subsequent Lanczos steps, $v_1$ participates in the orthogonalization so that the selective orthogonalization recommended by Parlett and Scott [363,353] is accomplished automatically.


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