In this lab you will define a struct type to represent complex numbers, create accessor functions to add, multiply, and find the magnitude of complex numbers, and then implement an interesting computation using complex numbers.
Getting Started
As always, you may refer to Lab 1 if you need a reminder about how to start the Cygwin Bash Shell or Notepad++.
Begin by downloading CS101_Lab21.zip. Save the zip file in the H:\CS101 directory.
Start the Cygwin Bash Shell and run the following commands:
cd h:
cd CS101
unzip CS101_Lab21.zip
cd CS101_Lab21
Start the Notepad++ text editor. Use it to open the files
H:\CS101\CS101_Lab21\Complex.cpp
When you are ready to compile the program, in the Cygwin window type the command
make
To run the program, in the Cygwin window type the command
./Complex.exe
Your Task
There are three tasks.
First Task
Define a struct data type called Complex. Instances of this type represent complex numbers consisting of a real component and an imaginary component.
You should add two fields of type double to the struct, one to represent the real part of the complex number, one to represent the imaginary part.
Second Task
At the top of Complex.cpp are six function prototypes declaring functions whose names begin with “complex_”. Implement these functions as follows:
struct Complex complex_create(double real, double imag)
Return a struct Complex instance that has the given real and imaginary values.
struct Complex complex_add(struct Complex c1, struct Complex c2)
Return the sum of the two complex numbers given as parameters. If a and b are the real and imaginary parts of c1, and c and d are the real and imaginary parts of c2, then the sum has a+c as its real part, and b+d as its imaginary part.
struct Complex complex_multiply(struct Complex c1, struct Complex c2)
Return the product of the two complex numbers given as parameters. If a and b are the real and imaginary parts of c1, and c and d are the real and imaginary parts of c2, then the product has ac-bd as its real part, and bc+ad as its imaginary part.
double complex_magnitude(struct Complex c)
Return the magnitude of the complex number given as the parameter. The magnitude is the square root of the sum of the squares of the real and imaginary parts.
double complex_get_real(struct Complex c)
Returns the real part of the complex number given as the parameter.
double complex_get_imaginary(struct Complex c)
Returns the imaginary part of the complex number given as the parameter.
Once you implement these functions, you can test your implementation by running the program:
(1) Run tests, or (2) Mandelbrot set? 1 complex_get_real (first test): passed complex_get_real (second test): passed complex_get_imaginary (first test): passed complex_get_imaginary (second test): passed complex_add (real part): passed complex_add (imaginary part): passed complex_multiply (real part): passed complex_multiply (imaginary part): passed complex_magnitude: passed All tests passed!
The last line of output should read All tests passed!
Third Task
The Mandelbrot Set is a famous mathematical object. It is a set of complex numbers defined by an equation with complex numbers Z and C:
Z = Z2 + C
The complex number C is in the Mandelbrot sert if and only if the equation can be iterated an infinite number of times without the magnitude of Z ever reaching 2. “Iterating” the equation works as follows:
- The initial value of Z is real=0, imaginary=0
- From the current value of Z, we can compute a new value of Z by squaring the old value and adding C
The third task is to complete the mandelbrot_num_iters function, whose prototype looks like this:
int mandelbrot_num_iters(struct Complex c, int max_iters);
It takes a complex number, which represents C in the equation above. The function should return the number of times the equation (Z = Z2 + C) can be iterated before the magnitude of Z reaches 2. If the number of iterations reaches max_iters, then max_iters should be returned: this caps the maximum number of iterations (because for points within the Mandelbrot set, the equation can be iterated an infinite number of times.)
Once you implement this function, you can run the program and enter the value 2 when prompted. You should then see something similar to the following (click for full-size image):
Note that you will get a more accurate picture if you make the font smaller and the terminal window larger.
What you are seeing is a graphical representation of part of the Mandelbrot set on the Complex plane. Each point is a complex number C, where the x coordinate is the real part, and the y coordinate is the imaginary part. Black locations are complex numbers that are in the set. The other colors represent how many iterations were required to show that a particular complex number was not in the set.
Submitting
When you are done, run the following command from the Cygwin bash shell:
make submit
You will be prompted for your Marmoset username and password, which you should have received by email. Note that your password will not appear on the screen.
Important:
You must submit your work before leaving class. If you do not submit work, you will not receive any credit for the lab.
