TABLE OF CONTENTS A single-step numerical method has a short memory. The only information passed from one step to the next is an estimate of the proper step size and, perhaps, the value of f(t n, y n) at the point the two steps have in common.
As the name implies, a multistep method has a longer memory. After an initial start-up phase, a pth-order multistep method saves up to perhaps a dozen values of the solution, y n ? p+1, y n ? p+2, , y n ?1, y n, and uses them all to compute y n+1. In fact, these methods can vary both the order, p, and the step size, h.
Multistep methods tend to be more efficient than single-step methods for problems with smooth solutions and high accuracy requirements. For example, the orbits of planets and deep space probes are computed with multistep methods.
