Suppose that an approximation 
to 
is computed
in an arithmetic of precision 
, where f is a real scalar
function of a real scalar variable. How should we measure the
``quality" of 
?
In many cases, we would be happy with a tiny relative error, 
, but this cannot always be achieved. Instead of
focusing on the relative error of 
we can ask ``for what
set of data have we actually solved our problem?", that is, for what 
do we have 
? In general, there may be
many such 
, so we should ask for the smallest one. The value
of 
, possibly divided by |x|, is called the
backward error. The absolute and relative errors of 
are
called forward errors. Figure 3 highlights the relationship
between these errors.
The process of bounding the backward error of a computed solution is called backward error analysis. It interprets rounding errors as being equivalent to perturbations in the data. The data frequently contains uncertainties due to previous computations or errors committed in storing numbers on the computer. If the backward error is no larger than these uncertainties then the computed solution can hardly be criticized.
A method for computing 
is called backward stable if,
for any x, it produces a computed 
with a small backward
error, that is, 
for some small
. The definition of ``small" is context dependent. Many
a times we use 
, where 
is the
machine epsilon.