C software for some new autonomous methods

This work is an extension of our previous paper [3], in which we have started introducing new C software for autonomous ordinary differential equation initial-value problems y' = f(y), y epsilon R-n, y(x(0)) = y(0), x(0) epsilon R, y(0) epsilon R-n, which implements new Runge-Kutta methods [1], [2]. The novel feature of this approach is the replacement of evaluations of f by approximations or evaluations of f(y). The advantage of this new method lies in the fact that fewer evaluations of fare required than in the standard Runge-Kutta methods and, usually, f(y) can be approximated to the desired accuracy with very little arithmetic. In effect, the new methods can be thought as multi-step Runge-Kutta methods. In this paper, we introduce some new Goeken-Johnson interpolation methods. We have also extended the capabilities of the software and we compare the classical Runge-Kutta methods of orders 3, 4 and 5, and the corresponding new Goeken-Johnson methods using both f(y) and approximations of f(y). We present numerical results of these comparisons. These results (for performance and accuracy) indicate that the new methods can be at least comparable if not better than the classical methods.