
\subsection{\bf Use of High Order Newton Cotes}

\begin{slide}

\begin{itemize}

\item One could also generate a formula in terms of the same function values
$f(x_i)$ by using the Newton-Cotes formula appropriate to $2n+1$
points.  

\item This gains an extra derivative in error just like Simpson, and
error is $ch^{2n+3}\,f^{2n+2} (\xi)$ where $c$ is some known number.

\item In practice, this is not often used because 

\begin{description}
\item{$\bullet$ (a)} the implied interpolation of $f(x)$ over a wide
range by a high order polynomial can produce disastrous results when
$f(x)$ not polynomial (e.g., has similarities in its value
$\underline{\rm or}$ derivatives).  

\item{$\bullet$ }On the other hand, local
interpolation by $f(x)$ by a low order polynomial (used in (NI.9)) is
much safer;

\item{$\bullet$ (b)} it is much easier to estimate error in iterated
Simpson of equation (NI.9) as we will describe in Section~NI.2.
\end{description}
\end{itemize}
\end{slide}

