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Error Bound for Computed Eigenvalues.
From (7.105) and the
perturbation theory of eigenvalues and eigenvectors, it can shown that
up to the first order of
residual norms and
,
there is an eigenvalue of satisfying

(212) 
where ``'' denotes ``less than'' up to the first order of
and
.
Note that
the individual condition number
is defined as
and and are the corresponding eigenvector and left
eigenvector of and are normalized so that
.
Since exact and usually are not known, for
all practical purposes, one could simply estimate this condition number
by
instead.
Notice that
, with equality if is Hermitian.
If the approximate left eigenvector is not available, bounds
directly on
do not exist in general.
The equation (7.104) would be the only thing available
to explain the accuracy of the
computed eigenvalues and .
Next: Error Bound for Computed
Up: Stability and Accuracy Assessments
Previous: Transfer Residual Errors to
Contents
Index
Susan Blackford
20001120