- One could also generate a formula in terms of the same function values
by using the Newton-Cotes formula appropriate to
**2n+1**points. - This gains an extra derivative in error just like Simpson, and
error is where
**c**is some known number. - In practice, this is not often used because
- (a) the implied interpolation of over a wide
range by a high order polynomial can produce disastrous results when
not polynomial (e.g., has similarities in its value
derivatives).
- On the other hand, local
interpolation by by a low order polynomial (used in (NI.9)) is
much safer;
- (b) it is much easier to estimate error in iterated
Simpson of equation (NI.9) as we will describe in Section NI.2.

- (a) the implied interpolation of over a wide
range by a high order polynomial can produce disastrous results when
not polynomial (e.g., has similarities in its value
derivatives).

Geoffrey Fox, Northeast Parallel Architectures Center at Syracuse University, gcf@npac.syr.edu