2.1 The Adams method
A typical numerical method for an initial value ODE system computes the solution on a step-by-step basis. Thus, the Euler method advances the solution from t0 to t1 using y0 as an initial value. Next, to advance from t1 to t2, we discard y0 and employ y1 as the new initial value.
Numerical analysts, however, are thrifty by nature. Why discard a potentially valuable vector y0? Or, with greater generality, why not make the solution depend on several past values, provided that these values are available?