1. 5.6 Multistep Methods
The methods discussed to this point in the chapter are called
one-step methods because the approximation for the mesh
point involves information from only one of the previous
mesh points, .
Although these methods might use functional evaluation information at points between and , they do not retain that information for direct use in future approximation.
All the information used by these methods is obtained within the subinterval over which the solution is being approximated.

2. multistep method
Method using the approximation at more than one previous
mesh point to determine the approximation at the next point
are called multistep method. The precise definition of these
methods follow, together with the definition of the two types
of multistep methods.

3. Definition 5.14
(5.22)
(5.23)

4. multistep method When the method is called explicit, or open, since
Eq. (5.23) then gives explicitly in terms of previously
determined values.
When the method is called implicit, or closed, since occurs on both sides of Eq. (5.23) and is specified only implicitly.

5. Example EXAMPLE The equations
for each , define an explicit four-step method known as
the fourth-order Adams-Bashforth technique .
EXAMPLE The equations
for each define an implicit three-step method known as the fourth-order Admas-Moulton technique.

6. local truncation error for multistep methods
The local truncation error for multistep methods is defined
analogously to that of one-step methods. As in the case of
one-step methods, the local truncation error provides a
measure of how the different equation fails to solve the
difference equation.

7. Definition 5.15
(5.31)

10. Examples To determine the local truncation error for the
three-step Adams-Bashforth method given as follows.

11. Examples Adams-Bashforth Two-Step Explicit Method:.
where The local truncation error is
for some

12. Examples Adams-Bashforth Four-step Explicit Method:.
where The local truncation error is
for some

13. Examples Adams-Moulton Two-step Implicit Method:.
where The local truncation error is
for some

14. Examples Adams-Moulton Three-step Implicit Method:.
where The local truncation error is
for some