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The partial generalized Schur form
can be obtained in a number of successive
steps. Suppose that we have the partial
generalized Schur form
and
. We want to
expand this partial generalized Schur form
with the new right Schur vector
and the left Schur vector
to
and
The new generalized Schur pair
satisfies
or, since
,
The vectors
and
can be computed from
Hence, the generalized Schur pair
is an
eigenpair of the deflated matrix pair
![\begin{displaymath}
\begin{array}{l}
\left(\left(I-{Z}_{k-1}{Z}_{k-1}^\ast\rig...
...ight)B
\left(I-Q_{k-1}Q_{k-1}^\ast\right)\right).
\end{array}\end{displaymath}](img2742.png) |
(228) |
This eigenproblem can be solved again with the
Jacobi-Davidson process that we have outlined in §8.4.1.
In that process we construct vectors
that are orthogonal to
and vectors
that are orthogonal to
. This simplifies the
computation of the interaction matrices
and
, associated with the
deflated operators:
![\begin{displaymath}
\left\{\begin{array}{l}
M^A\equiv W^\ast\left(I-{Z}_{k-1}{...
...\ast\right) V=W^\ast B V,\rule{0pt}{3.5ex}
\end{array}\right.
\end{displaymath}](img2748.png) |
(229) |
and
and
can be simply computed as
and
,
respectively.
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Susan Blackford
2000-11-20