An alternative way of improving the Euler formula rests on the Taylor
expansion of the solution of the initial value problem. By retaining
the first two terms in the expansion of 
about 
we obtain
the Euler formula, so by keeping more terms we should obtain a more accurate
formula. Assuming that f has continuous second partial derivatives, so that
has at least three continuous derivatives in the interval of interest,
we can write
where 
is some point in the interval 
.
Since
By differentiting this equation and then setting 
we find that
The three-term Taylor formula is obtained by replacing 
by its
approximate value 
in the formulas for 
and
, and then neglecting the term 
in Eq. (1). Thus we obtain
Assuming temporarily that 
, it follows that the local
truncation error 
associated with formula (2) is
where 
. Thus the local truncation error
for the three-term Taylor formula is proportional to 
, just as
for the improved Euler formula discussed earlier. Again, it can also be shown
that for a finite interval the global truncation error is no greater than a
constant times 
.
The three-term Taylor formula requires the computation of 
and
, and then the evaluation of these functions as well as
at 
. In some problems 
and 
may be more
complicated than f itself. If this is the case, it is probably better to
use a formula with comparable accuracy, such as the improved Euler formula,
that does not require the partial derivatives 
and 
.
In principle, four-term or higher order Taylor formulas can be developed.
However, such formulas involve even higher partial derivatives of f and are
in general rather awkward to use.
Finally, we note that if f is linear in both t and y, as in problem in the example we have been using, then the improved Euler and three-term Taylor formulas are identical.
