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Stopping Criteria and Accuracy Assessment
Since the eigenproblem of a symmetric indefinite pencil
does not have any special
mathematical properties, we may refer to techniques developed for general
nonHermitian matrices to assess the quality of the approximate eigentriplets
of an indefinite pencil. Nonetheless, we can take advantage of the symmetry
to simplify the analysis and subsequently the error bound. The simplification
stems from the simple relation between the right and left eigenvectors of
. If is a right eigenvector of ,
then is the corresponding left eigenvector.
To assess the accuracy of a Ritz triplet
,
by the discussion in §7.8,
it is known that there is a matrix such that
where
In fact,
is the optimal backward error.
By the standard firstorder perturbation expansion or the error
bound presented in §7.13, there is an
eigenvalue of , such that
where we have made use of equations (8.24), (8.25), and the
equality
Furthermore, to avoid a bound which explicitly involves the Ritz
vectors , we note that
and
where it is assumed that
.
Recall that the columns of are normalized to 1.
The value of
may be updated in each Lanczos iteration.
Therefore the error
can also
be bounded by
which does not explicitly use the Ritz vector .
In conclusion, one may use

(239) 
as a provisional error estimate in the inner loop of
Algorithm 8.4 (i.e., step (19)).
Only when the required number of Ritz
values
have passed this test, and not before,
may the Ritz vectors be computed. At that point the more
precise factor,
may be computed at the extra cost of forming .
The Ritz values
approximate the eigenvalues of ,
while the quantities
are approximations to the eigenvalues
of the original problem (8.21). We can estimate
using the bound above and
Next: Singular
Up: Symmetric Indefinite Lanczos Method
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Susan Blackford
20001120