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Transfer Residual Error to Backward Error.
It can be proved that there are Hermitian matrices and , e.g.,

(104) 
such that
and are an exact eigenvalue and
its corresponding eigenvector of .
We are interested in such matrices with smallest possible norms.
It turns out that the best possible
for the
spectral norm
and the best possible
for Frobenius norm
satisfy

(105) 
See [256,431,473].^{}In fact, is given explicitly by (5.40).
Therefore if is small,
the computed
and are exact ones
of nearby matrices. Error analysis of
this kind is called backward error analysis and
matrices are backward errors.
We say an algorithm
that delivers an approximate eigenpair
is
backward stable for the pair with respect to the norm
if it is an exact eigenpair for with
.
With these in mind,
statements can be made about the backward stability of the algorithm which
computes the eigenpair
.
In convention, an algorithm is called backward stable
if
.
Next: Error Bound for Computed
Up: Some Combination of and
Previous: Residual Vector.
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Susan Blackford
20001120