Next: Error Bounds for Computed
Up: Positive Definite
Previous: Residual Vector.
Contents
Index
Transfer Residual Error to Backward Error.
It can be proved that there are Hermitian matrices , e.g.,

(93) 
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 the best possible for the spectral norm
and the best possible for the Frobenius norm
satisfy

(94) 
See [256,431,473].
In fact, is given explicitly by (5.29).
So 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
, where
is the machine precision.
Next: Error Bounds for Computed
Up: Positive Definite
Previous: Residual Vector.
Contents
Index
Susan Blackford
20001120