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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 clustered eigenvalues,
as guaranteed by (4.54), 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 (4.56).
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, starting with
a residual matrix

where is diagonal with diagonal entries consisting of
approximations to all the eigenvalues in the cluster, and the columns of
are the corresponding approximate eigenvectors.
Assume that has orthonormal columns and that is
closer to , the diagonal matrix whose diagonal entries
consist of all the eigenvalues in the cluster. Let be the eigenvector
matrix associated with , and let be the smallest difference
between any approximate eigenvalue in the diagonal of
and those eigenvalues of not presented in the diagonal of .
Then [101]

where
is diagonal with diagonal entries being
the arccosines of the singular values of .
Because of the way it is defined, this gap is expected
to be big and thus
will be small as
long as is.

** Next:** Remarks on Eigenvalue Computations
** Up:** Stability and Accuracy Assessments
** Previous:** Error Bound for Computed
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Susan Blackford
2000-11-20