2.3.1 The Adams-Bashforth and Adams-Moulton Formulas     continued...

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.