We begin with numerical methods for solving a scalar version of (10), i.e., the case for n=1:
The methods we develop for solving (17) can easily be extended to systems of first order differential equations and to higher order differential equations. The methods are referred to as discrete variable methods and generate a sequence of approximate values for , at points . No attempt is made to approximate the exact solution, , over a continuous range of the independent variable t. In our development, we will assume a constant spacing h between t points. In realistic implementations of these methods, however, h is chosen to satisfy a user-specified accuracy request. The expression will always be used to denote the solution to (17) at , and will always be used for an approximation to .
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.