TABLE OF CONTENTS The popular methods before the advent of modern digital computers for the step-by-step integration of differential equations had an essential feature in common. At each step of the process, use was made of the function values already obtained in the previous steps. Thus, if we had arrived at the value y n, then to determine y n+1 these methods required the use of the values for y n-1, y n-2, , the number of which depended on the desired accuracy and on the particular method employed. They were based on simple finite-difference formulas which were easy to apply using manual methods. However, the disadvantages were that they required special start-up procedures and were not readily amenable to changing the size of the integration interval.
The Runge-Kutta methods do not utilize preceding function values and so were frequently used by hand computers for starting an integration process. Then, the switch was made to finite-difference methods because the Runge-Kutta formulas were too difficult to continue by hand. However, in programming a method for the digital computer, it is inconvenient to use special instructions for a starting process. Furthermore, constant shifting of data is required which is difficult and time-consuming in a digital program the manual computer operator does this simply by moving his eyes down the page. On the other hand, the more complicated formulas of the Runge-Kutta methods are easily programmed and are, today, frequently preferred to the more complex logic required for...
