%Monte Carlo Integration
%If you modify this file, please indicate here and in the footnote
%Math263B

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\Large
Monte Carlo Integration\footnote{Copyright \copyright 2002 Larry Snyder and Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}
%modified by 2006 by Todd Young}
\end{center}

\begin{enumerate}
\item Type \verb& help rand & and read the first paragraph of resulting help page.
      Next enter the following:\\
      \verb&  x = rand(10,1)  &\\
      \verb&  y = x.^3        &\\
      \verb&  avg = sum(y)/10 &\\
      Figure out what happens in these command before you proceed. 
\item Enter the following sequence commands:
   \begin{enumerate}
      \item  \verb&n = 10&
      \item  \verb&x = rand(n,1); avg=sum(x.^3)/n&
      \item  Use the $\uparrow$ key to recall this line
             and then press \verb&Enter& again.
      \item  Obtain 10 estimates this way and record the values
             you get along with the absolute error of each estimate.
             You can have \textsc{MatLab} calculate the absolute error for
             you conveniently by including at the end of line of \#1(b): 
             \verb& error = abs(.25 - avg)&.
      \item Explain why this is an approximation of 
	$\int_0^1 x^3  \, dx$. (It has to do with the average of a function.)
   \end{enumerate}

\item\label{exp}  Enter the command \verb& n = 100 & and  use the
      $\uparrow$ key to recall the line in \#1(b) again.  Press the
      enter key to execute this line.  Obtain and 
      record 10 estimates this way along with the absolute errors.

\item Repeat this process using \verb& n = 1000 & and \verb& n = 10000&.

\item Make a chart showing the relationship between the sample size 
      \verb&n& and the arithmetic mean of the absolute errors of 
      the estimates with sample size \textsf{n}. The data should reflect
      the relationship $|E_n| \approx Kn^{-r}$. Use the data and
      logarithms to determine the constants $K$ and $r$.

\item For the Trapezoid rule \verb$r = 2$ and for Simpson's rule \verb$r = 4$.
      How does the random method introduced here compare with the Trapezoid and
      Simpson's methods of numerical integration? Which is the most accurate, which
      the least? 

\item Prepare a brief written report answering all
      the questions. Use complete sentences and standard mathematical notation.
      Do {\bf not} get a printout.

\end{enumerate}

\vfill
\noindent
\textsf{This demonstrates the connection between averages and integrals. 
Because this technique is efficient in 
higher dimensions, variants (known as Monte Carlo methods) are actually used 
in practice to evaluate integrals.}

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