 CS-381 -- Lab 5: Monte Carlo Integration

Fall Quarter 1999

Electrical Engineering and Computer Science Department
Dr. Christopher C. Taylor

S-331, 277-7339 www.msoe.edu/~taylor/

### Purpose

The purpose of this lab assignment is to gain an appreciation of how statistical techniques can be used to solve deterministic problems.

### Background

One common type of random number generator is the linear congruential random number generator (LCRNG). It may be defined recursively as:

xn+1 = (a xn + c) mod m.

The ability of the random number generator is highly dependent on the constants a, c, and m and the initial condition x0 (aka, the seed).

### Assignment

In this lab, you are to do the following:

• Implement a linear congruential random number generator.
• Experiment with different values for the constants. In one of your tests, use a=16807, c=0, m=231-1, and x0=667790.
• Speculate on the role of the different constants. How does the choice of each change the characteristics of the numbers generated?
• Use the generalized Monte Carlo method described in class to integrate the following functions:
• f(x) = sin(x), for x in [0, ]
• f(x) = -4x/(1 + x2)2, for x in [-1.5,1.5]
• f(x) = (2/(1 + x2)) - x2, for x in [-1,1]
You should calculate these integrals using your LCRNG and cstdlib's rand() function. Be sure to comment on any differences.
• 10% of your grade for this assignment will be based on the design of your program. A simple object-oriented program design is desirable.

### Formatting Notes

• When discussing the results of your experiments with the linear congruential random number generator, you may find it useful to include graphs. These graphs should be included as separate files and follow these guidelines:
• Each file should begin with your initials, for example: tcFig1.pdf
• Each graph should be in one of the following formats and should end with the appropriate suffix: .pdf, .eps, .png, or .jpg
• Include the graphs within your report using the following command:
    \begin{center}
\includegraphics[width=.5\textwidth]{tcFig1}
\end{center}

• The three f(x)'s may be included in your report using the following commands:
$f(x) = \sin(x), \qquad \mbox{ for } x \in [0, \pi]$
$f(x) = \frac{-4x}{(1 + x^2)^2} \qquad \mbox{ for } x \in [-1.5, 1.5]$
$f(x) = \frac{2}{1 + x^2} - x^2 \qquad \mbox{ for } x \in [-1, 1]$

• The update equation for the LC random number generator can be written as:
  $x_{n+1} = (a x_n + c) \mbox{ mod } m$


For this project, you may work individually or with a partner.

### Lab report (due midnight Tuesday, October 26, 1999)

This week, your lab report need not be self-contained. If you worked with a partner, only one report is required for the two of you. Your report should include:

• Discussion containing:
• A brief description of any problems you encountered and how they were resolved
• A brief description of your design plan