What would happen if we were to increase the order even more? For p=3 and p=4 the formulas become
and we obtain using the results that appear in Table 3. Note that the same order of accuracy is obtained for p = 4 with as for p=2 and . Of course, higher total derivatives are easy to compute for .
Table 3: Approximate solution using the Taylor series algorithms of order 3 and 5.
Taylor series methods can be quite effective if the total derivatives of f are not too difficult to evaluate. Software packages are available that perform exact differentiation, facilitating their use (ADIFOR, MAPLE, MATHEMATICA, etc.). However, most of today's software packages for solving IVPs do not employ Taylor series methods.
(See exercise set 3.)