A nonlinear element function 
is assumed to be a function of the
problem variables 
, a subset of the overall variables x.
Suppose that has 
components. Then one can consider
the nonlinear element function
to be of the structural form 
, where we assign 
. The elemental variables
for the element function are the variables v and, while we
need to associate the particular values with v, it is
the elemental variables which are important in defining the character
of the nonlinear element functions.
As an example, the first nonlinear element function for a particular problem might be
which has the structural form
where we need to assign 
and 
.
For this example, there are three elemental variables.
The example may be used to illustrate a further point. Although 
is a function of three variables, the function itself is really only
composed of two independent parts; the product of 
with 
, or, if we write 
and 
, the product of 
with 
. The
variables and 
are known as internal variables
for the element function. They are obtained as linear
combinations of the elemental variables. The important feature as
far as an optimization procedure is concerned is that each nonlinear
function involves as few internal variables as possible, as this
allows for compact storage and more efficient derivative
approximation.
It frequently happens, however, that a function does not have useful internal variables. For instance, another element function might have structural form
where for example 
and 
. Here, we have
broken 
down into as few pieces as possible. Although there are
internal variables, 
and 
, they are the same in
this case as the elemental variables
and there is no virtue in
exploiting them. Moreover it can happen that although there are
special internal variables,
there are just as many internal as
elemental variables and it therefore doesn't particularly help to
exploit them. For instance, if
where, for example, 
and 
, the function can
be formed as 
, where 
and 
. But as there are just as many internal variables as elementals,
it will not normally be advantageous to use this internal
representation.
Finally, although an element function
may have useful internal
variables, the user may decide not to exploit them. The optimization
procedure should still work but at the expense of extra storage
and computational effort.
In general, there will be a linear transformation from the elemental variables to the internal ones. For example in (2.6), we have
while in (2.7), we have
In general the transformation will be of the form
and this transformation is useful if the matrix W has fewer rows than columns.
