A major limitation of Runge-Kutta formulas is the amount of
work required; work is measured in terms of the number of
times the function **f** is evaluated. For higher order formulas,
the work goes up dramatically; **p** evaluations per step lead to
procedures of order **p** for **p=1, 2, 3**, and 4, but not for 5; 6
evaluations are required for a formula of order 5, 7 for
order 6, 9 for order 7, 11 for order 8, etc. For this reason,
fourth order procedures are quite common. As in the second order
case where the parameter was arbitrary, there is a family of
fourth order formulas that depend on several parameters. One choice
leads to the so-called * classical* formulas.

To obtain an approximate solution of order **p = 4** to the IVP (17) on
, let and generate the sequences

where

and , .

As we pointed out earlier, the methods developed extend readily to
systems of first order IVPs of the form (10). As an illustration, the
formulas (35) in the case of an **n**-dimensional system become

where