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It is possible to design an iterative algorithm for which
or 
is not directly available,
although this is not the case for any algorithms in this book.
For completeness, however, we discuss stopping criteria in this case.
For example, if ones ``splits'' 
to get the iterative
method 
, then the
natural residual to compute is
.
In other words, the residual 
is the same as the residual of the
preconditioned system 
. In this case, it is
hard to interpret 
as a backward error for the original system
, so we may instead derive a forward error bound
.
Using this as a stopping criterion requires an estimate of
.
In the case of methods based on
splitting , we have 
,
and 
.
Another example is an implementation of the preconditioned conjugate
gradient algorithm which computes 
instead of 
(the
implementation in this book computes the latter). Such an
implementation could use the stopping criterion 
as in Criterion 5. We
may also use it to get the forward error bound 
, which could also
be used in a stopping criterion.