Adams-Bashforth & Adams-Moulton Methods

We have seen that the exact solution of the differential equation satisfies the identity

for any two points and in the interval . In the methods discussed earlier, we tried to replace the function , which is unknown, by an interpolating polynomial having the values on a set of points . The Newton backward difference formula was used to find such an interpolating polynomial. We assume here that the interpolating points and has a continuous the derivative. Using the notations that and

where is the backward difference operator with we have the interpolating polynomial, for the unknown function , with the remainder term as