UPPER BOUNDS ON THE COMPUTATIONAL COMPLEXITY OF ORDINARY DIFFERENTIAL EQUATION INITIAL VALUE PROBLEMS

Abstract

With few exceptions, past work in analytic complexity theory has centered on the problem of finding the zero of a nonlinear scalar function or operator. In this paper, we consider the problem of finding upper bounds on the number of function evaluations sufficient to solve a system of ordinary differential equations to within a given error criterion, 6, with one-step and multistep methods. Our main results are as follows:(1)For any 6 there is a unique choice of order and step size which minimizes the number of function evaluations.(2)As 6 decreases, this “optimal order” and the number of function evaluations both increase. As 6 →0, both the optimal order and the number of function evaluations tend to infinity, but very slowly.(3)As 6 → 0, the optimal order for multistep methods is lees than the optimal order for one-step methods; moreover, numerical results indicate that the optimal multistep order becomes less than the optimal multistep order within a practical range of interest.(4)As 6 → 0, the cost of the optimal multistep method is greater than the cost of the optimal one-step method; however, numerical results indicate that the optimal multistep method is cheaper for all 6 within a practical range of interest.