A differential equation has no ``memory''. That is, the values of for
**t** before do not directly affect the values of for **t** after
. Some numerical methods have memory, and some do not. We shall
first describe a class of methods known as * one-step* methods. They
have no memory; given there is a recipe for that depends
only on information at .

Suppose we want to approximate the solution to (17)
on the interval .
Let the **t** points be equally spaced; so for some positive integer **n** and
, , . If **a<b**, **h** is
positive and we are integrating forward; if **a>b**, **h** is negative and we
are integrating backwards. The latter case could occur if we were solving
for the initial point of a solution curve given the terminal point. A
general one-step method can then be written in the form

where is a function that characterizes our method. We seek accurate algorithms of the form (21). By this we mean algorithms for which the true solution, , almost satisfies (21), i.e.,

with ``small.'' The quantity is called the
* local (truncation) error* of the method. The method (21) is said
to be of * order* **p** if for all , and for
all sufficiently small **h**, there are constants **C** and **p** such that