Two Theorems on the Accuracy of Numerical Solutions of Systems of Ordinary Differential Equations

In this paper we present some results concerning the accuracy with which a numerical solution of the system of ordinary differential equations x + J(x, 0 = 0 , t ^ 0 [x(0) = xo] (1) can be obtained by the use of a numerical integration formula of the well-known type 1 V V bky'n-k , n ^ p. (2) jt = - l In (2) the y n are approximations to the x n = x(nh), where h, a positive number, is the step size parameter; y0, y x , · · · , y v are starting vectors, the last p of which are obtained by an independent method; and yn+i = UkVn-k + h k-Q y'n = - j ( y n nh). If i v^ 0, then yn+ is defined implicitly, and (2) is said to be of closed type. It is assumed throughout t h a t (2) can be solved* for yn+1 for all n ^ p. Specializations of (2) include, for example, Euler's method: , yn+1 = Vn + hy'n , and the more useful formula yn+1 = yn + h(y'n + y'n+l). It is assumed throughout t h a t for t ^ 0, f(x, t) is a well-defined real iV-vector-valued function defined in the set of all real iV-vectors x, t h a t f{x, t) satisfies (the usual weak) conditions which guarantee the existence and uniqueness of a solution of (1), and t h a t the Jacobian matrix df(x, t)/dx exists for all x and all t ^ 0. Equation (2) ignores the roundoff error R n introduced in calculating yn+i, and, in order to take R n into account, we shall consider instead * It is well known t h a t if / satisfies a uniform Lipshitz condition, and if h is sufficiently small, then (2) possesses a unique solution yn+i which can be obtained by a simple iterative process.