Commerce: Financial Risk Management
CS 222
Project 2 – Array/String, Structure, Recursion, and Makefile
This project is to practice using array, structure, recursion, and multiple components compilation. It is an extension to the previous homework.
The Mandelbrot Set
The Mandelbrot set is a set of all complex numbers c that satisfy the following condition: the absolute value of mandelbrot(n) remains bounded when n (a natural number 0, 1, 2, …) tends to infinity, where
mandelbrot(n) = mandelbrot(n-1)*mandelbrot(n-1) + c;mandelbrot(0) = c.
Here the addition and multiplication are complex number addition and multiplication.
Specifically, for our project, if the absolute value of mandelbrot(15), |mandelbrot(15)|, is smaller than 10000 for a given c, then c is in the Mandelbrot set. If we consider a c as a 2D coordinates on a plane, the points in the rectangular area from lower-left corner (-2.0, -1.12) to upper-right corner (0.47, 1.12) will contain the Mandelbrot set. Our goal is to calculate a 2D array of points that covers the Mandelbrot set in this area and display the 2D array as an image. If a point is in the Mandelbrot set, we display a 1 (or #); otherwise, we display a 0 (or an empty space). You may save the area in a file so the area may contain many more points beyond your display. The number of points in the array, in other words, the number of row and columns, is up to you, but 40*30 will be sufficient to demonstrate the image.
Implementation details
1) You may use more files if you want, but here I recommend having complex.h, complex.c, Mandelbrot.h, Mandelbrot.c, Mandelbrot_print.h, Mandelbrot_print.c, main.c, and makefile. As always, you should write comments and implement the project a few statements at a time.
2) Constants, structure definitions, global variables, and function prototypes are defined in the header files for inclusions. Here in “complex.h”, you should have structure type definition for “complex_t” and function prototypes for “add_complex”, “multiply_complex”, and “abs_complex”.
3) In “complex.c”, you should include “complex.h”, so to have corresponding structure definition, and implementations of the functions. In general, related functions are grouped/implemented in the same c file. In “complex.c”, implement all the functions related to complex number operations as defined in the prototypes in the header file.
If you have system include files, they should be placed before the user defined header files. In general, never put a header file in another header file. They should be included explicitly.
4) mandelbrot.c implements the recursive function “complex_t mandelbrot(int n)”. In general, you should avoid using global variables, so the recursive function has to carry another input value, c, and mandelbrot(c, n) checks whether c belongs to the Mandelbrot set. In other words, your function should be mandelbrot(complex_t c, int n) instead. Optionally, for the purpose of learning, you may define a global variable c in main.c, and use “extern” in mandelbrot.h to make it visible in mandelbrot.c, so you can just use mandelbrot(n).
mandelbrot(n) uses complex number addition and multiplication, so you should include “complex.h” in mandelbrot.c to allow mandelbrot(n) to access complex number functions. In the recursive function, |mandelbrot(n)| can be quite large beyond the computer’s representation. Therefore, in the recursion, if |mandelbrot(n-1)| is bigger than 10000, your recursion should consider that it is not bounded already, so you return mandelbrot(n) = (10000, 10000) instead of the calculated number to indicate that current c is not in the Mandelbrot set. Also, in the recursion, mandelbrot(n-1) is calculated twice, which is really wasting computing effort, so you should just calculate it once for efficiency.
5) In order for main.c to see the recursive function, you need to include “mandelbrot.h”. Because you use complex numbers in main.c, you need to include “complex.h” in main.c as well. Because mandelbrot function uses structure, so you should have “complex.h” included ahead of “mandelbrot.h”. Also, in order to print the generated image, you need to include “Mandelbrot_print.h”.
In main.c, for all point c in the area from (-2.0, -1.12) to (0.47, 1.12), we can check the corresponding mandelbrot(15) to see if it is in the Mandelbrot set. If it is in the set, which means that |mandelbrot(15)| is smaller than 10000, we save a ‘1’ in a 2D character array; otherwise, we save a ‘0’. At the end of each row, we save a NULL to make it a string. Therefore, we create an image of the Mandelbrot set saved in the 2D array. For example, you can check 40*30 points with a double for-loop in a rectangular area. That is, we start in x direction from -2.0 with step size (0.47-(-2))/40 = 0.06175 to 0.47, and in y direction from -1.12 with step size (1.12-(-1.12))/30 = 0.077 to 1.12. After that, we call Mandelbrot_print with the input of the 2D array pointer and the number of rows to print the image.
6) in Mandelbrot_print.c, you print the image by rows of strings. The function prototype is in Mandelbrot_print.h. If you want, as an option, which is not required for this project, you may save the image in a file, so you can check out a much larger image in the file with many more points. You can google images on Mandelbrot, and see the Mandelbrot set images with very fine points at the screen resolution with vivid colors.
7) In Makefile, the linking is usually the first rule in the Makefile. You need to link complex.o, Mandelbrot.o, Mandelbrot_print.o, and main.o into an executable called Mandelbrot. You should have rules that compile complex.c, Mandelbrot.c, Mandelbrot_print.c, and main.c into object files separately. Make sure the object files depend on the corresponding header files as well. In other words, for example, if you modify complex.h, complex.c, Mandelbrot.c, and main.c will be compiled again if you type “make”.
Testing and Submitting
Test your program on Mason to make sure that it compiles with “make” and runs properly. While on Mason, create a script file listing your programs to the screen in the following orders: Makefile, mandelbrot.h, mandelbrot.c, complex.h, complex.c, Mandelbrot_print.h, Mandelbrot.c, and main.c. Demonstrate a sample run with the Mandelbrot image. Submit all files (except the objects and the executable) to Blackboard.
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