where 
is the solution of the initial value problem. The basic idea
of an Adams method is to approximate 
by a polynomial
of degree k - 1 and to use the polynomial to evaluate the
integral on the right side of Eq. (1). The coefficients in
are determined by using previously calculated data
points. For example, suppose that we wish to use only the points
and 
. Then the polynomial 
is
of degree one and has the form 
. Since 
is to be an
approximation to 
, we require that 
and that 
. Recall that we denote
by 
for an integer j. Then A and B must satisfy
the equations
Solving for A and B, we obtain
Replacing 
by 
and evaluating the integral
in Eq. (1), we find that
Finally, we replace 
and 
by 
and
, respectively, and carry out some algebraic simplification.
For a constant step size h we obtain
Equation (2) is the second order Adams-Bashforth formula.
It is an explicit formula for 
in terms of 
and 
and has a local truncation error proportional to 
.
We note in passing that the first order Adams-Bashforth formula, based on the
polynomial 
of degree zero, is just the original
Euler formula.
More accurate Adams formulas can be obtained by following the procedure
outlined above, but using a higher degree polynomial and correspondingly more
data points. For example, suppose that a polynomial 
of degree
three is used. The coefficients are determined from the four points
and 
. Substituting this polynomial for
in Eq. (1), evaluating the integral, and simplifying the
result, we eventually obtain the fourth order Adams-Bashforth formula,
namely,
The local truncation error of this fourth order formula is proportional
to 
.
A variation on the derivation of the Adams-Bashforth formulas gives another set
of formulas called the Adams-Moulton formulas. To see the difference, let
us again consider the second order case. Again we use a first degree
polynomial 
, but we determine the coefficients by
using the points 
and 
.
Thus 
and 
must satisfy
and it follows that
Substituting 
for 
in Eq. (1) and simplifying,
we obtain
which is the second order Adams-Moulton formula. We have written
in the last term to emphasize that the
Adams-Moulton formula is implicit, rather
than explicit, since the unknown 
appears on both sides of the
equation. The local truncation error for the second order Adams-Moulton
formula is proportional to 
.
The first order Adams-Moulton formula is just the backward Euler formula, as you might anticipate by analogy with the first order Adams-Bashforth formula.
More accurate higher order formulas can be obtained by using an approximating
polynomial of higher degree. The fourth order Adams-Moulton formula, with a
local truncation error porportional to 
, is
Observe that this is also an implicit formula because 
appears in
.
Although both the Adams-Bashforth and Adams-Moulton formulas of the same order have local truncation errors proportional to the same power of h, the Adams-Moulton formulas of moderate order are in fact considerably more accurate. For example, for the fourth order formulas (3) and (4), the proportionality constant for the Adams-Moulton formula is less than 1/10 of the proportionality constant for the Adams-Bashforth formula. Thus the question arises: should one use the explicit (and faster) Adams-Bashforth formula, or the more accurate but implicit (and slower) Adams-Moulton formula? The answer depends on whether by using the more accurate formula one can increase the step size, and therefore reduce the number of steps enough to compensate for the additional computations required at each step.
In fact, numerical analysts have attempted to achieve both simplicity and
accuracy by combining the two formulas in what is called a
predictor-corrector method. Once 
, and
are known we can compute 
, and 
,
and then use the Adams-Bashforth (predictor) formula (3) to
obtain a first value for 
. Then we compute 
and use the
Adams-Moulton (corrector) formula (4), which is no longer implicit,
to obtain an improved value of 
. We can, of course, continue to
use the corrector formula (4) if the
change in 
is too large. However, if it is necessary to use the
corrector formula more than once or perhaps twice, it means that the step size
h is too large and should be reduced.
In order to use any of the multistep methods it is necessary first to
calculate a few 
by some other method. For example, the fourth order
Adams-Moulton method requires values for 
and 
, while the fourth
order Adams-Bashforth method also requires a value for 
. One way to
proceed is to use a one-step method of comparable accuracy to calculate the
necessary starting values. Thus, for a fourth order multistep method, one
might use the fourth order Runge-Kutta method to calculate the
starting values.
Another approach is to use a low order method with a very small h to
calculate 
and then to increase gradually both the order and the step
size until enough starting values have been determined.
