Runge-Kutta methods with minimum error bounds

Introduction. Numerical methods for the solution of ordinary differential equations may be put in two categories-numerical integration (e.g., predictorcorrector) methods and Runge-Kutta methods. The advantages of the latter are that they are self-starting and easy to program for digital computers but neither of these reasons is very compelling when library subroutines can be written to handle systems of ordinary differential equations. Thus, the greater accuracy and the error-estimating ability of predictor-corrector methods make them desirable for systems of any complexity. However, when predictor-corrector methods are used, Runge-Kutta methods still find application in starting the computation and in changing the interval of integration. If, then, Runge-Kutta methods are considered in the context of using them for starting and for changing the interval, matters such as stability 2, 3 and minimization of roundoff errors 4 are not significant. Also, simplifying the coefficients so that the computation will be speeded up is not important and, on modern computers, minimization of storage 4 is seldom important. In fact, the only criterion of significance in judging Runge-Kutta methods in this context is minimization of truncation error. It is the purpose of this paper to derive Runge-Kutta methods of second, third and fourth order which have minimum truncation error bounds of a specified type. We will consider only the case of integrating a single first-order differential equation because this is the only tractable case analytically. But it seems reasonable to assume that methods which are best in a truncation error sense for one equation will be at least nearly best for systems of equations. 2. The General Equations. For the solution of the equation