 
Summary: MULTISTEP METHODS
All of the methods we have studied until now are ex
amples of onestep methods. These methods use one
past value of the numerical solution, call it yn, in order
to compute a new value yn+1. Multistep methods use
more than one past value, and these are often more
e cient than the earlier methods. For example,
yn+1 = yn+
h
2
[3f(xn; yn) f(xn 1; yn 1)] ; n 1
(1)
is an explicit, stable, second order method with trun
cation error of size O h3 . We derive it below.
Multistep methods can be derived in a number of
ways, but the most popular methods are derived most
easily using numerical interpolation and numerical in
tegration.
Integrate the equation
Y 0(x) = f (x; Y (x)) (2)
