Computational complexity of one-step methods for systems of differential equations

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 p*h-order Taylor series method requiring only 0(p^ In p) arithmetic operations per step as p -» +a>. (Moreover, we show that any such algorithm requires at least O(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 t decreases, tending to infinity as i 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. This research was supported in part by the National Science Foundation under Grant MCS75-222-55 and the Office of Naval Research under Contract N00014-76-C-0370, NR 044-422.