Consider the dot product of two vectors, 
, where
. Since the order of evaluation of
affects the analysis (but not the
final error bound), we will assume that the evaluation is from left
to right.
Let 
denote the ith partial sum. Using
the IEEE error model, we have
where 
. For our analysis we will not
differentiate between the different 
, so to simplify the
expressions let us drop the subscripts on the 
and write
. Then
Based on this pattern, we have
There are various ways to simplify this result. In particular, we make use of the following lemma:
Lemma
If
where
and 
for 
, and 
, then
Applying the lemma to 
, we obtain
This is backward error results and may be interpreted as follows: the
computed inner product is the exact one for a perturbed set of data
Each perturbation is ``small".