Last Update 29 October, 2003

## Computer-aided Mathematics (Toshiyuki Sugawa)

### Sample Programs

Simple programs designed for Mathematica (Notebook style file)
A program producing animation of orbits under a quadratic polynomial
A program producing the Julia set of a quadratic polynomial
Web resources of programs drawing Julia sets and the Mandelbrodt set
Microscope for Mandelbrodt set
Other than the above, you will find various Java programs on the web if you check by a search engine.

### Exercises

Choose at least two exercises from the list below and make reports on it. (If you choose the marked one for experts, then you may choose nothing else.)
Exercise 1. Modify the second sample program for Mathematica in the above into that of producing the Mandelbrodt set and actually give pictures of the whole and a very small part of the Mandelbrodt set.
Exercise 2. Improve the above sample programs and make interesting, colorful or beautiful pictures of various Julia sets.
Exercise 3. Produce a picture of the Julia set of a rational function which appears in Newton's method to solve a cubic equation. (Hint: for a polynomial P(z), the above rational funtion is given by f(z)=z-P(z)/P'(z). The corresponding Julia set can be obtained by coloring the point according to the root of f(z) to which the orbit approaches.
Exercise 4. Make pictures of Julia sets of polynomials with degree greater than two or rational functions.
Exercise 5 (for experts). Code a program of drawing Julia sets or the Mandelbrodt set for Java Applet so that the user can choose parameters of the quadratic polynomial, and put the Java Applet program on a suitable web page. Also, make some comments on the source code so that one can understand the structure of the program.

### How to post your reports

Please choose one of the following. (Upper is more preferrable.)
1. Construct a web page containing your reports and notify Sugawa of the URL.
2. Email to Sugawa a reporting document written in MS Word or Adobe PDF formats.
3. Print out your reports in a suitable form and bring it to the office of Math. Dept.
The deadline is the end of November, 2003. If you have a special reason for the delay, you may claim, not later than 25th of November, to Sugawa about the reason and the date until which you will submit the reports. Note that there is no guarantee that your claim will be accepted. Note also that you must give the source code and some explanation about it.