It is easy to vectorize the formation of 
since
each element 
can be written as
independent of 
apart from multiplication by a factor
. The vectorized algorithm for the
evaluation of , adapted from reference [17],
appears in Figure 5. An analogous algorithm
can be used to obtain 
. However, we can also obtain
for 
states by
as proven by Olsen and co-workers [17].
Figure 7: Vectorized Algorithm for .
A fairly straightforward algorithm for evaluating 
is
presented in Figure 6. The vectorized
algorithm for the evaluation of is presented in
Figure 7. This algorithm makes use of a
gather and scatter operation to avoid indirect addressing.
Figure 6: Simple Algorithm for .
Figure: Vectorized Algorithm for .
For , an improvement to the algorithm can be made
by utilizing an identity similar to equation (6.31).
The ijklth component of is related to the klijth
component via
