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Error Bounds for Computed Eigenvalues.
Rewrite as
By (4.54) on p. , we have

(95) 
for some eigenvalue of the pair .
A good estimate to is needed to use this bound.
With more information, a much better bound
can be obtained.
Let us assume that
is
an approximation of the eigenpair of the pair.
The ``best'' corresponding to
is the Rayleigh quotient
,
so we assume that has this value.
Suppose that is closer
to than any other eigenvalues of the pair, and
let be the gap between
and any other eigenvalue of the pair. Then

(96) 
This improves (5.31) if the gap is
reasonably big. In practice we can always pick the better one.
This bound also needs information on
, besides the residual error and .
Usually such information
is available after a successful computation by,
e.g., the shiftandinvert Lanczos
algorithm, which usually delivers eigenvalues in the neighborhood
of a shift and consequently yields good information on the
. This comment also applies to the bounds in
(5.33) and (5.34)
below.
Next: Error Bounds for Computed
Up: Positive Definite
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Susan Blackford
20001120