Let us consider the case of having
an approximate eigenpair
to an exact pair
. Let
be the Schur decomposition of
(see §2.5, p.
),
where
is unitary and
For the small and dense eigenproblems,
can be efficiently estimated. This is available in LAPACK [12].
However, for large sparse eigenproblems, since
is generally
not available, the estimation of
is out of the question. We can only get a gross sense
about the quality of computed eigenvectors.
Note that
roughly measures
the separation of
from the eigenvalues of
. We
have to say ``roughly'' because
As summarized in [198],
the separation of the eigenvalues
has a bearing upon eigenvector sensitivity. Indeed,
if is a nondefective, repeated eigenvalue, then there
are an infinite number of possible eigenvector bases
for the associated invariant subspace. The preceding analysis merely
indicates that this indeterminacy begins to be felt as
the eigenvalues coalesce.
It is well known that each individual eigenvector associated with
eigenvalue clusters is very sensitive to perturbations, and consequently
such an eigenvector cannot be accurately computed in general.
Fortunately, often for the case of an eigenvalue cluster,
it is the entire associated eigenspace that is of practical importance, and
that eigenspace can be computed with satisfactory accuracy.
Detailed treatment is beyond the
scope of this book and the interested reader is referred to the
literature, for example, [425].