### 2.2.2 Runge-Kutta Methods     continued...

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