When a spectral transformation is used, additional considerations
should be made with respect to stopping criteria to take advantage of
the special nature of the transformed operator . Moreover, the quality
of the approximate eigenvectors can be improved significantly with a minor
amount of postprocessing. In order to compute eigenvalues of
near to
we will compute the eigenvalues
of
that are of largest magnitude. It is not really necessary for
to
be extremely close to a desired eigenvalue. However, for the following
discussion it is worth keeping in mind that in practice, it is typical
to take
near to
desired eigenvalues, and hence it is typical for
to hold.
Let with
, where
. Since
A simple rearrangement of (7.23) gives
Adding and subtracting
on the right-hand side of (7.24) and rearranging terms will
result in
A heuristic argument further supports the use of instead of
.
From (7.23) it follows that
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This vector has not yet been scaled to have unit norm. However,
,
so the error bound will decrease after
is normalized.
Moreover, if
,
then the floating point computation of the norm will already result
in
without any rescaling.
From (7.23), the residual
is orthogonal to the Krylov space spanned by the
columns of
. However, this Galerkin condition is lost upon transforming
the computed eigenpair to the original system,
regardless of whether we use
or
. This is because
is not the Rayleigh quotient
associated with
or
. However, from (7.25)
On the other hand, from
(7.24) we deduce that
Equations (7.25) and (7.27) imply
that the vector is a better approximation than
to the
eigenvector associated with the approximate eigenvalue
provided that
is greater than 1. Moreover,
when
, only a moderately small Ritz estimate is needed to
achieve an acceptably small direct residual and Rayleigh quotient error.
If the
is near the desired eigenvalues, then these
eigenvalues are mapped by
to large eigenvalues and typically
.
The above analysis is based on that given by Ericsson and Ruhe [162] for the generalized symmetric definite eigenvalue problem.