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Transfer Residual Errors to Backward Errors.
It can be shown that the computed
eigenvalue and eigenvector(s) are the exact ones of a nearby matrix pair,
i.e.,
and
if is available,
where error matrices and are small relative to the
norms of and .
 Only is available but is not. Then
the optimal error matrix (in both the 2norm
and the Frobenius norm) for which
and are
an exact eigenvalue and its corresponding eigenvector of the pair
satisfies

(241) 
 Both and are available. Then
the optimal error matrices (in the 2norm) and
(in the Frobenius norm) for which
,
, and are
an exact eigenvalue and its corresponding eigenvectors of
the pair satisfy

(242) 
and

(243) 
See [256,431,473].
We say the algorithm
that delivers the approximate eigenpair
is
backward stable
for the pair with respect to the norm
if it is an exact eigenpair for
with
; analogously
the algorithm that delivers the eigentriplet
is backward stable for the triplet with respect to the norm
if it is an exact eigentriplet for
with
. With these in mind,
statements can be made about the backward stability of the algorithm which
computes the eigenpair
or
the eigentriplet
.
Conventionally, an algorithm is called backward stable
if
.
Next: Error Bound for Computed
Up: Stability and Accuracy Assessments
Previous: Residual Vectors.
Contents
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Susan Blackford
20001120