With more information, a better error bound can be obtained.
Let us assume that
is
an approximation of the eigenpair
of
.
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
, and
let
be the gap between
and any other eigenvalue:
.
Then we have
Note that (4.55) needs information on
, besides the residual error
.
Usually such information
is available after a successful computation by,
e.g., a Lanczos algorithm
with SI,
which usually delivers eigenvalues in the neighborhood
of a shift and consequently yields good information on
. This comment also applies to the bound in (4.56)
below.