With more information, a better error bound can be obtained.
Let us assume that
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.