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In the case when the eigenvalue
has one or more other eigenvalues of close by, in other words,
when belongs to a cluster of eigenvalues,
as guaranteed by (5.31) the computed
is still accurate as long as
is tiny, but the computed
eigenvector may be inaccurate because of the
appearance of the gap in the denominator of
(5.33). It turns out that each individual
eigenvector associated with the clustered eigenvalues
is very sensitive to perturbations,
but the eigenspace spanned by all the eigenvectors associated with
the clustered eigenvalues is not. Thus for the clustered eigenvalues,
we should instead compute the entire eigenspace.
A theory along the lines given above can be established to
show that the difference between the computed eigenspace and the
eigenspace associated with the cluster is inversely proportional to the
gap defined as the smallest difference between
any eigenvalue in the cluster and any other eigenvalue not
in the cluster. Because of the way it is defined, this gap is expected
to be big.

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Susan Blackford
2000-11-20