One way to guarantee accuracy in the solution of an I.V.P. is to solve the problem twice using step sizes h and and compare answers at the mesh points corresponding to the larger step size. But this requires a significant amount of computation for the smaller step size and must be repeated if it is determined that the agreement is not good enough. The Runge-Kutta-Fehlberg method (denoted RKF45) is one way to try to resolve this problem. It has a procedure to determine if the proper step size h is being used. At each step, two different approximations for the solution are made and compared. If the two answers are in close agreement, the approximation is accepted. If the two answers do not agree to a specified accuracy, the step size is reduced. If the answers agree to more significant digits than required, the step size is increased. Each Runge-Kutta-Fehlberg step requires the use of the following six values: Then an approximation to the solution of the I.V.P.
The methods of Euler, Heun, Taylor and Runge-Kutta are called single-step methods because they use only the information from one previous point to compute the successive point, that is, only the initial point is used to compute and in general is needed to compute . After several points have been found it is feasible to use several prior points in the calculation. The Adams-Bashforth-Moulton method uses in the calculation of . This method is not self-starting; four initial points , , , and must be given in advance in order to generate the points . A desirable feature of a multistep method is that the local truncation error (L. T. E.) can be determined and a correction term can be included, which improves the accuracy of the answer at each step. Also, it is possible to determine if the step size is small enough to obtain an accurate value for , yet large enough so that unnecessary and time-consuming calculations are eliminated. If the code for the subroutine is f