Reply to comment by L.-S. Yao and D. Hughes

Theauthors make some strong statements, not merely about the system that they call and that I shall call eq. (1), but regarding the general practice of approximating solutions of ordinary differential equations (ODEs) numerically. Before assessing these statements one must distinguish between just plain convergence and uniform convergence. Let Xn(t), n = 1, 2, . . ., be a sequence of vector functions of time t and let X(t) be another vector function. Of special interest here are cases where X is a particular true solution of a system of ODEs, where there exists a sequence of time increments τ n approaching 0 as n → ∞, and where the functions Xn are approximations to X produced by procedures that are identical except that the nth approximation uses the increment τ n. The sequence Xn converges to X at time t′ if, given any ε > 0, there exists a corresponding N(t′) such that |Xn(t ′) − X(t ′)| < ε if n > N (t ′).