Complex Variables- $15.95
- 313 pages
- Schaum's Outlines
- 1983
- ISBN: 0070602301
- Order from Amazon
- $80.31
- 1995
- 386 pages
- ISBN: 0079121470
- Order from Amazon
The Mandelbrot Set is defined by a simple iterative function:
zt+1 = zt2 + c
z0 = 0
This is an iterative equation because the value of
zt depends on the value of zt-1.
Given the values of z0 it seems trivial to write
a computer program that calculates zt for
arbitrary t. In fact matters are a little more complicated
than they seem at first. In particular, both
z and c
are complex numbers with both real and imaginary parts.
Furthermore, depending on the value of c, this equation
may exhibit sensitive dependence on initial conditions
and become chaotic.
The Mandelbrot set is defined as those points in the complex plane for which this iteration does not diverge (reach infinity; i.e. grow arbitrarily large). It can be shown that if the absolute value of this iteration exceeds 2, then it diverges.
Complex VariablesYou may also want to search the Web for more info. There are many existing Mandelbrot set programs on the Web, many available with source, including at least one by your professor. While it may be educational to investigate these, I very much suspect that none of them would actually be an answer to the specific problems described here.
Use full-length variable names where plausible.
Indent code with spaces rather than tabs.
Do not overdo it. This is a straight-forward and simple problem that requires a straight-forward and simple answer. It can be solved in a single method, without defining more than one class, using arrays or exceptions, or accessing any parts of the class library we have not yet discussed.
Include your name, email address, homework number (1), and a brief description of the problem in header comments. Include other comments as necessary.
Write a program that reads c and the number of generations from the command line, and prints that many iterations of the Mandelbrot equation. Also print (at the beginning of the output) the value used for c.
Be sure to handle all user errors (e.g. not providing two values on the command line) to the extent possible. Check the ranges of the data the user enters (e.g. don't let the number of generations be less than 0). Print an error message and exit if a value outside the allowed range is entered. Provide a default value if none is given.
Write a program that reads c and the number of generations from the command line, and prints that many iterations of the Mandelbrot equation. Also print (at the beginning of the output) the value used for c.
Be sure to handle all user errors (e.g. not providing two values on the command line) to the extent possible. Check the ranges of the data the user enters (e.g. don't let the number of generations be less than 0). Print an error message and exit if a value outside the allowed range is entered. Provide a default value if none is given.
Detect if divergence has occurred (absolute value greater than 2) and stop output if so. Note that this test does not require you to know about or use any square root, power, exponent, or other mathematical functions that Java may or may not have.
Write a ComplexNumber class which provides:
Make the interface to this class as natural and similar to the
existing numeric primitive data types as possible. In particular,
you may want to investigate the interfaces of the
java.math.BigInteger and
java.math.BigDecimal classes.
Rewrite Problem 3 using this ComplexNumber
class.
Write a MandelbrotUtilities class with two
methods:
ComplexNumber c
and a number of generations n, and returns the value
of the nth iteration of the Mandelbrot equation for that
value of c.ComplexNumber c
and a number of generations n, and returns the number
of generations required for divergence, (i.e. for an absolute value
of 2 to be reached) or -1 if the equation does not diverge within
the specified number of generations.Write a simple command line user interface for this class in a different class. This should allow the user to input a value of c and a number of generations. It should tell the user whether or not the equation diverged, and if it did diverge, how many generations were required to prove this.
Add equals(), hashCode(), and
toString() methods to the ComplexNumber class.
Be sure to think carefully about what it means for two complex
numbers to be equal and document and justify your choice.
Add a method that returns the absolute value of a complex number
to the ComplexNumber class.
Rewrite Problem 5 to take advantage of this new method.
Make the ComplexNumber class cloneable.
Place the ComplexNumber class in a suitably named
package. Then rewrite Problem 5 to use this new class. However,
leave the Problem 5 classes in the default package.
Write a version of the ComplexNumber class that uses
java.math.BigDecimal instead of doubles internally.
(Does it make sense to provide overloaded versions of the public
methods that take big decimals or strings as arguments instead of
doubles?) Place it in the same package used above. Then rewrite
Problem 5 to use this new class. However, leave the problem 5
classes in the default package.
Write an abstract superclass for complex numbers of which the two previous problem classes can be subclasses. Make them subclasses of this new superclass. Put it in the same package as above. Then rewrite the problem 5 classes to use this new class with a user option to choose double or arbitrary precision.
Rewrite the user interface class(es) from previous problems to properly handle the case where the user enters a value for c or the number of generations that is not recognizable as a number.
To the complex number class add a valueOf()
method modeled after the valueOf() methods of the
java.lang.Number subclasses
such as java.lang.Double that converts
a String to a ComplexNumber.
Begin by writing at least ten different examples of various forms you will
recognize as a complex number and at least ten different examples of
things you will not recognize as a complex number.
Rewrite your user interface class to take advantage of this method when parsing the command line arguments.
Using the MandelbrotUtilities class from problem 5,
write an applet that draws a graph of the absolute value of |z(n)|
vs. n where z(n) is the nth iteration of the Mandelbrot equation
for a fixed c. (You only need to draw the graph. You do not need to
draw axes or legends or anything of that nature.) Let the y axis
run from 0 to 2.
Use PARAM tags to specify the c and the number of
iterations to track.
Handle all exceptions reasonably. (Printing an error message on
System.out and exiting is not reasonable.) Assume one
generation per horizontal pixel. Make this applet reasonably
quick.
Make the applet available on a Web page. Include at least three instances of the applet on the page: one with a converging c, one with a diverging c, and one with a c that neither converges nor diverges.
Write an applet that draws a scatter plot of successive positions of z(n) where z(n) is the nth iteration of the Mandelbrot equation for a fixed c. (You only need to draw the graph. You do not need to draw axes or legends or anything of that nature.) Let the y axis run from -2 to 2 and the x-axis from -2 to 2.
Use PARAM tags to specify c and the number of
iterations to track.
Handle all exceptions reasonably. (Printing an error message on
System.out and exiting is not reasonable.) Assume one
generation per horizontal pixel. Make this applet reasonably
quick.
Make the applet available on a Web page. Include at least three instances of the applet on the page: one with a converging c, one with a diverging c, and one with a c that nither converges nor diverges.
To the applets in Problems 13 and 14, add the necessary user interface components to enable the user to set c and the number of generations.
Use the necessary layout managers to arrange the applet.
Write a Mandelbrot image producer. Use this to write an applet that draws the entire Mandelbrot set.
Write a subclass of either Canvas or
Component that draws the Mandelbrot set using a Mandelbrot
image producer. Rewrite Problem 17 to use this class instead of
drawing the graph directly.
Add text fields that allow the user to set the x range and the y range.
Make last class's applet a panel. Write an applet that uses the
panel. Make the panel and the applet that contains it reasonably
attractive by using a LayoutManager.
Then write a frame based stand-alone application that uses frames, each containing an instance of the above panel. Don't do anything that would prevent the program from being expanded to a multiple window application.
Hand in the application as a runnable jar archive on a 3.5in floppy disk. Please put the floppy in an envelope and tape the envelope to the print out of your source code and screen shots of the running program. Label the floppy disk and make sure the label is firmly affixed to the floppy and will not get caught in the grader's floppy disk drive.
Add alert dialogs that warn the user when they're attempting to set an invalid value.
Add File and Edit menus to the frame of last week's program. The File menu should contain New, Open, Close, Save, Save As, Page Setup, Print, and Exit menu items. The Edit Menu should contain Undo, Cut, Copy, Paste, and Clear menu items. All menu items should have the standard menu shortcuts. The New, Close, and Exit menu items should work. The rest should be disabled. Use menu separators and ... where customary.
Make Save, Save As, and Open all work. You will need to design a file format for the data.
Write a thread safe version of the complex number class.
Use a thread pool with ten threads to calculate multiple points simultaneously.
Make printing work.
Make Copy and Paste work.
Expand this entire assignment so it works for arbitrary Julia sets.