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cedure for the initial-value problem of ordinary differential equation : (E) (IV) Many numerical analysts have been investigating the discrete variable methods for the problem. Consequently everyone can enjoy to solve numerically ordinary differential equation in almost all computing centers. It seems as if we got the numerical integrator through the use of the computer. But the study is yet continued for "better" numerical procedure. Among the one-step methods, Runge-Kutta methods (RK methods, in short) are popular because of the high accuracy and the feasibility of changing step-size. In general the methods are expressed as follows. The solution of (E) at x 0 + h is approximated by

Runge-Kutta type integration formulas including the evaluation of the second derivative. I