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 and
for , and , then
where
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".