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2.2 One-Step Methods

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