
\subsection{\bf NI.5:\ \ Monte Carlo Integration}

\begin{slide}
\begin{itemize}
\item (i) ``All'' integration methods can be written as
$$
I = \int^b_a g(x)\,f(x)\,dx = \sum^n_{i=0} w_i f\left( x_i\right)
$$
\item where the $n+1$ points $x_i$ and $n+1$ weights $w_i$ are
determined in sundry ways to optimize various properties, as
described earlier.  

\item If we increase $n$, the dependence of error is
$$\vbox{\halign{#\hfil\tabskip=2em&#\hfil&#\hfil\tabskip=0pt\cr
Error $\propto$ & ${1\over{n^2}}$ & iterated Trapezoidal\cr
&&\cr
& ${1\over{n^4}}$ & iterated Simpson\cr
&&\cr
& ${1\over{n^{2m}}}$ & iterated $m$-order Gauss\cr
}}$$
\item with iterated groups of $k$ points.
\vspace{10pt}

\centerline{\psfig{figure=ni.fig6.ps,clip=true,width=5.0in}}
\end{itemize}
\end{slide}

