

\subsection{\bf Double Monte Carlo's Again --- I}

\begin{slide}
\begin{itemize}
\item A digression on double integrals, which we already discussed in
NI.4 in general terms
$$
I = \int f(x)\,dx \int g(x,y)\,dy
$$
\item Each integral $\int$ could be several dimensions, but write as one.  If
\begin{eqnarray*}
\int f(x)\,dx & = & I_1\ \ \ \mbox{needs}\ N_1\ \mbox{points} \\
\int g(x,y)\,dy & = & I_2\ \ \ \mbox{needs}\ N_2\ \mbox{points} \\
\end{eqnarray*}
\item Then for analytic method (such as Romberg), the calculation of $I$ would
need (approximately) $N_1\cdot N_2$ points.  Computer time is
``multiplicative''.  
\item What about Monte Carlo!  Fractional error
$$
{\delta I_i\over{I_i}} = \left( {\sigma_i\over{I_i}}\right) /
\sqrt{N_i}
$$
\end{itemize}
\end{slide}

