A too small value of **h** means that we are doing unnecessary computation
that could lead to roundoff error (error induced because the arithmetic is
not exact); a too large value of **h** means that we are probably not
meeting the desired accuracy requirements (errors induced by discretization).
Although we will not develop the details here, something can be said about the
qualitative behavior of global error as a function of **h** for a one-step
method of order **p**. If the error in evaluating is
with for all **i**
and the error
in forming is with for
all **i**, then it can be shown that

where **C** and **L** are constants depending on **y**
and its derivatives. A
graph of (37) has the qualitative behavior shown in Figure 3.

Figure 4: Qualitative behavior of absolute error
at a fixed point as a function of the step size

As we said earlier, most codes estimate local error at each step and attempt
to adjust **h** accordingly. A widely used procedure for estimating local
error follows. Suppose we have computed approximations of order **p** and **p+1**
at , and respectively. Then using
(20),
the local error in the **p**th order approximation can be written as

If **h** is sufficiently small, the term
can be neglected and we can use the computed value as an estimate of the error in the **p**th order formula.
This sort of approximation has been validated by extensive numerical
experimentation over the years.