The methods are referred to as discrete variable methods and generate a sequence of approximate values for , at points . In our development, we will assume a constant spacing between points. In realistic implementations of these methods, however, is chosen to satisfy a user-specified accuracy request.
Errors enter into the numerical solution of IVPs from two sources.
The first is discretization error and depends on the method being used. The second is computational error which includes such things as roundoff error , the error in evaluating implicit formulas, etc. In general, roundoff error can be controlled by carrying enough significant figures in the computation. The control of other computational errors again depends on the method being used. There are two measures of discretization error commonly used in discussing the accuracy of numerical methods for solving IVPs.