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.