On the attainable order of Runge-Kutta methods

If the elements of f are differentiable arbitrarily often with respect to x and the elements of y, the true solution and the numerical solution can be expanded in powers of h. It has been shown [1] that these two expansions agree up to terms in h' (that is, the process is of order p) if (1) 4= l/y whenever r < p. In this formula, b is a typical elementary weight, r is its order, and y is a certain integral constant associated with it. 'F itself is a polynomial of degree 1 in bl, b2, * * *, b, and degree r -1 in all , a12 , * * a,, . To find the values of y for the different 'Z, one may use a result proved in [1]. If 4i, , ,2 * have values yi, 72, ... , y corresponding to them and if (2) cI [4'142 ... then