Computational Complexity of One - Step Methods for Systems of Differential Equations * By Arthur

The problem is to calculate an approximate solution of an initial value problem for an autonomous system of N ordinary differential equations. Using fast power series techniques, we exhibit an algorithm for the pth-order Taylor series method reN quiring only 0(p In p) arithmetic operations per step as p —> + °°. (Moreover, we N show that any such algorithm requires at least 0(p ) operations per step.) We compute the order which minimizes the complexity bounds for Taylor series and linear Runge-Kutta methods and show that in all cases this optimal order increases as the error criterion e decreases, tending to infinity as e tends to zero. Finally, we show that if certain derivatives are easy to evaluate, then Taylor series methods are asymptotically better than linear Runge-Kutta methods for problems of small dimension N.