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.

- 2.2.1 Taylor Series Methods
- 2.2.2 Runge-Kutta Methods
- 2.2.3 Some Implementation Issues
- 2.2.4 The Codes RKSUITE
- 2.2.5 RKSUITE Example