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2.2 One-Step Methods     continued...

This can be interpreted as meaning that goes to zero no slower than . Hereafter, we shall write terms like this as . The constant C depends, in general, on the solution , its derivatives, and the length of the interval over which the solution is to be found, but is independent of h.

Note that, the order of our method is p even though the order of the local (truncation) error is p+1, because these errors tend to accumulate as the integration proceeds. The order of a method may be viewed as a measure of how fast the error in the computed solution goes to zero at a fixed point t as more and more steps are taken, i.e., as h approaches zero. Our goal is to find functions that are inexpensive to evaluate, yet of as high order p as possible. In what follows, different functions are displayed, giving rise to the Taylor series methods and the Runge-Kutta methods.