CSCI 323 Modeling & Simulation

Project #1: Sensitive Dependence on Initial Conditions

Date Due: 21 September 2009

As you might remember, there are actually THREE cube roots of 1: (If you don’t believe it, cube each one and see what you get.) We are going to use Newton’s method to find the cube roots of 1, starting with various complex numbers, and determining to which of the three roots that starting value converges. Depending on which value it is, we will color that point differently. For example, if the starting point converges to the first value, color that point red (0xff0000); if it converges to the second value, color that point green (0x00ff00); if it converges to the third root, color it blue (0x0000ff).

In case you don’t remember your Newton’s method, it’s fairly simple. We are trying to find the zeroes of a function, and we make a starting guess. We plug that value into the function, do the computation, and see what value we get. If it’s not zero, we evaluate the derivative of the function at that point to get the slope, and plot the tangent line. Hopefully, the intersection of the tangent line with the horizontal line whose value is zero is a better approximation of the root. We iterate this process until we get as close as we want to the root. This process usually converges fairly rapidly.

For real functions, we would say , since we want value(s) for x that when cubed are equal to 1 (the zeroes of the function). For a value , the equation of the tangent line at that point is , so for , we get (assuming ). In general, the formula is just . This is the way you do it for real arithmetic…for complex arithmetic it’s the same basic equation, but you have to use the complex operators.

Before running your program, predict what behavior you think you will see. You might think it would look something like this:

(taking into account my lack of artistic talent), and you might be right…or you might be surprised too.

Your program should ask the user how many pieces to divide the real and imaginary axes into, and then plot that many points in the range from -1-i to 1+i. You should run your program, capture the output as a jpeg image, and mail me your source code and the image in a zip file. Since this is not a graphics course, here is some sample code that should help get you started.