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Transfer Residual Error to Backward Error.
There are Hermitian matrices
such that and are an exact eigenvalue and
its corresponding eigenvector of , i.e.,
One such is

(61) 
We are interested in such matrices with smallest possible norms.
It turns out the best possible for the spectral norm
and the best possible for Frobenius norm satisfy

(62) 
see, e.g., [256,431].
In fact, is given explicitly by (4.52).
So if is small, the computed and
are an exact eigenpair 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 this definition in mind,
statements can be made about the numerical stability of the algorithm which
computes the eigenpair
.
By convention, an algorithm is called backward stable
if
, where is the machine
precision.
Next: Error Bounds for Computed
Up: Stability and Accuracy Assessments
Previous: Residual Vector.
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Susan Blackford
20001120