The lowest order Adams-Bashforth formula arises from interpolating the single value by . The interpolating polynomial is constant so its integration from to results in and the first order Adams-Bashforth formula (AB1):

This is just the familiar forward Euler formula. For constant step size
**h**, the second order Adams-Bashforth formula (AB2) is also easily found
to be

The implicit Adams-Moulton formulas arises when the polynomial interpolates for :

When **j = p-1**, the right hand side contains the term , and we see that is defined only implicitly
by this formula. The solution is accomplished by first ``predicting'' the
result using the explicit Adams-Bashforth formula (42),
and then ``correcting''
it using the implicit formula (46); we then proceed by ``simple'' or
``functional'' iteration. If **L** is a bound on
and the step size **h** is small enough
so that for some constant ,

then (46) has a unique solution
and the error is decreased by a
factor of at each iteration. For ``small'' step sizes **h**, the
iteration converges very quickly.