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#### Problem statement overview

###### Computer Science

Problem statement overview.

In this lab assignment, your task is to write a C++ program that implements a complex number class. You will populate the complex number class with class functions that add,

1subtract, multiply, divide, square and take the absolute value of the complex numbers.

In the second part of the assignment, you will use your complex number class to compute whether or not each element of an array of complex numbers is inside or outside of a set called the Mandelbrot set. A number is in or out of the Mandelbrot set depending on whether thenumber converges or diverges when iterating the equation z= z2+c. You will then plot the

Mandelbrot set using ascii characters.

Put all functions with more than ≥ 3 code statements into the separate source code file

rather than coding them into the class.

2 Complex number class

2.1 Defining the Complex class

Create a class called Complex that has as variables a pair of coordinates x and y. Make these

of type double and private: Create the following standard functions for your class:

1. Accessor functions to return each of your variables

2. Mutator functions to set your two variables, individually and jointly.

3. Default constructor initializes values to zero.

4. Parameterized constructor initializes the values to given parameters, use the mutator

to do the setting.

5. Printing function. Overload this function to print both with and without an input

variable name.

You also want to be able to perform mathematics, so write functions:

1. void add(Complex c) : adds the coordinate c to the internal coordinate.

2. void sub(Complex c) : subtracts the coordinate c from the internal coordinate.

3. void mult(Complex c) : multiplies the complex number c with the internal complex

number

4. void div(Complex c) : multiplies the inverse of complex number c−1 with the internal

complex number

5. void invert() : inverts the internal complex number.6. void square(): squares the internal complex number

7. double abs(): returns the absolute value of the internal complex number.

Then create the corresponding overloaded operators. Note that these functions return type

Complex.

1. Complex operator + (Complex c) : the equivalent overloaded + operator, have it call

the add() function on an appropriate variable to do the job.

2. Complex operator - (Complex c) : the equivalent overloaded - operator, have it call

the sub() function on an appropriate variable to do the job.

3. Complex operator * (Complex c) : the equivalent overloaded multiplication * operator, that multiplies the two numbers and returns the value. Use the mult() function

on an appropriate variable to do the job.

4. Complex operator / (Complex c) : the equivalent overloaded division / operator,

that divides the two complex numbers and returns the complex value. Use the div()

function on an appropriate variable to do the job.

5. Complex operator ^ (int n) : an integer power operation that multiplies the internal complex number by itself n times and returns the value. Use the mul() function in

an appropriate way to do the job.

Some of these functions may be longer than 3 lines and their definitions should thereby go into

the .cpp file. Test your Complex class functions in a function testComplexClass() that you

call from the main() as we have been doing in the practicals. Ensure all the above-described

functionality is working properly in your test function.

2.2 Background on Complex numbers

In general, complex numbers are made up of 2 parts: the real part x and the imaginary part y.

It can be written as z = x+iy or (x,y) where i = √−1 is the imaginary number. Addition of

complex numbers can be done component-wise: (x0+iy0)+(x1+iy1) = (x0+x1)+i(y0+y1),

as can subtraction: (x0 + iy0) − (x1 + iy1) = (x0 − x1) + i(y0 − y1). Multiplication of two

complex numbers is implemented as:

(x0 + iy0)(x1 + iy1) = (x0x1 − y0y1) + i(x0y1 + x1y0).

Complex division is implemented as:

(x0 + iy0)

(x1 + iy1) =

(x0 + iy0)(x1 − iy1)

(x1 + iy1)(x1 − iy1) =

(x0x1 + y0y1) + i(x1y0 − x0y1)

x2 1 + y2 1In particular, the inverse of a complex number is given as:

1

(x0 + iy0) =

(x0 − iy0)

(x0 − iy0)(x0 + iy0) =

(x0 − iy0)

x2

0 + y2 0

The absolute value of a complex number is given as: abs(x+iy) = qx2 + y2. These complex

number mathematics are to be implemented into your complex number derived class.

In general, while real numbers can be conceptualized as a point on the real-number line,

complex numbers are conceptualized as a point on the complex-plane with the real component

lying along the x-axis and the imaginary component lying along the y-axis. In the next part

on the Mandelbrot set we will be testing complex numbers in the complex plane to see if

they meet the criteria to fall within the Mandelbrot set.

3 The Mandelbrot Set (MBS)

The Mandelbrot set is one of the most well known fractals consisting of a set of complex

numbers x+iy in the complex-number plane. There is a specific rule for testing whether a

complex number is included in Mandelbrot set or not that is described in Section 3.2.

3.1 Defining the Mandelbrot class

Define a Mandelbrot class in which you will test a large number of points in the complex

plane to check whether or not each point falls into the Mandelbrot set. For this purpose,

you have to define a set of points (x, y) with ranges defined by your Mandelbrot class. Thus

your Mandelbrot class should start out with the following private fields:

? double xMin, xStep, xMax; // The range of x-elements in the MBS

? double yMin, yStep, yMax; // The range of y-elements in the MBS

that hold the minimum, step and maximum values for both the x and y ranges of your

Mandelbrot set. Then the x-values to be tested are {xMin, xMin+xStep,...,xMax} and the

y-values to be tested are {yMin, yMin+yStep,...,yMax}. You also need variables keeping

track of the number of x and y-values that you have as this will define the size of the array

used to hold the Mandelbrot set. Define the number of points in the x and y ranges:

? int Nx, Ny; // Number of x and y elements in each dimension of the MBS

You will need to write a function to compute the values of Nx and Ny from xMin, xStep,

xMax and yMin, yStep, yMax respectively.The third private field you will need to define is a 2D array holding values of 1 or 0: a

1 indicates the corresponding point is in the Mandelbrot set while 0 indicates the point is

not in the set. The easiest way to implement this is as a 1D array indexed as 2D. Add a 1D

array (indexed as 2D) that you will dynamically allocate, to your class:

? int *MBS; // array representing the 2D MBS.

When you initialize your Mandelbrot object with values for xMin, xStep, xMax, yMin,

yStep, yMax, the constructor should subsequently calculate the Nx and Ny values and then

initialize your array int* MBS to have Nx*Ny values using:

MBS = new int[Nx*Ny];

and you need free the corresponding memory with delete[]MBS; in the destructor.

Create the following:

1. A default constructor to initialize your parameters to some sane startup values. You can

use xMin=-2;xStep=.02;xMax=1;yMin=-1;yStep=.02;yMax=1; as your default values.

Nx and Ny should be calculated and the required memory should be allocated in MBS.

2. A parameterized constructor where the input parameters are taken from the user,

calculates the MBS size and allocates the memory.

3. A destructor that frees the memory in MBS.

4. A function void calcNxNy() to compute the values of Nx and Ny

5. A function void printRange() to print out the range of x, y, Nx and Ny parameters

for the user to validate correctness.

6. A function (not a constructor) void setRange(double xMin, double xStep, double

xMax, double yMin, double yStep, double yMax) that takes the values of xMin,

xStep, xMax, yMin, yStep, yMax, saves them and recomputes Nx, Ny and then frees

the old memory and reallocates new memory for MBS. This function may be called if

the user ever wants to change the size or range of the Mandelbrot set being explored.

In both constructors, the values of Nx and Ny should be computed and the memory to hold

Nx×Ny ints should be dynamically allocated.

At this point you are ready to compute the Mandelbrot set.3.2 Determining the Mandelbrot set

The Mandelbrot set is determined by the complex-number equation/iteration:

z = z2 + c (1)

Equation (1) is iterated (repeated) over and over again starting with z=0, if the value of

abs(z) diverges (grows → ∞), then c is NOT in the Mandelbrot set. Whereas if z converges,

or oscillates between several points without diverging, then the complex number c IS in the

Mandelbrot set. To clarify this definition of the Mandelbrot set, let us check the complex

number c = 0 as an example:

? z=0 → z=z2+c = 0

? z=0 → z=z2+c = 0

and no matter how many times we iterate, the value of z never diverges, and therefore c = 0

IS in the Mandelbrot set and the corresponding value of the MBS array should be set to 1. If

we try with c = 1

? z=0 → z=02+c = 1

? z=1 → z=12+c = 2

? z=2 → z=22+c = 5

? z=5 → z=52+c = 26

and here we see that after a few iterations equation (1) diverges (|z| → ∞)) for c = 1.

Therefore c = 1 is NOT in the Mandelbrot set and the corresponding value of the MBS array

should be set to 0. A complex number c is inside or outside of the Mandelbrot set depending

on whether the iteration z = z2+c converges or diverges. You must come up with the test for

determining the convergence or divergence of a complex number c yourself using the Complex

class functions you have defined above.

You are to write the class function calcMBS() (together with supporting sub-functions)

that, for the given values of xMin, xStep, xMax, yMin, yStep, yMax, compute whether

each point is or is not in the Mandelbrot set and populates the 2D Nx*Ny array int *MBS

accordingly.

3.3 Printing the Mandelbrot set

After you have populated your array int *MBS with the calcMBS() function, you need to

write the printMBS() function to print out the 2D array to get a picture that looks as shown

in Figure 1Mandelbrot set range:

x:[-2:0.02:1], Nx=151

y:[-1:0.02:1], Ny=101

In function calcMBS()

In function printMBS()

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Figure 1: The output of the Mandelbrot set program with x ranging from -2 to +1 in steps of

0.02 and y ranging from -1 to +1 in steps of 0.02. A one indicates that complex number is part

of the MBS, while a zero indicates it is not part of the MBS.

4 Report

Write up a report (510 pages) detailing and documenting what you have done in this

assignment. You should particularly highlight in your report, the codes that implement the

above-described features. Have your report focus on the parts of the assignment that you

had to create for yourself, such as:

2. Details in how you wrote the function calcNxNy() function.

3. How you wrote the function to recalculate the the ranges and reallocate the memory

for the MBS.

4. How you determined whether equation (1) converges or diverges5. How you wrote the calcMBS() function to iterate over each point in the range and

determined whether the point is inside or outside of the MBS. Explain the logic behind

the design/creation of this function.

5 Submission and Evaluation

In order to standardize the evaluation of your work submit the following on Moodle by the

due date/time:

1. main.cpp file: the main program. This program calls the high-level test functions for

the Complex and Mandelbrot classes.

2. CW4.cpp file: contains the source code for all the functions used by your program

to achieve the objectives. Be sure to use clear comments to delineate the different

the functions/operators for the Complex class, and where are the functions/operators

for the Mandelbrot class.

3. CW4.h file: the header file that holds both the function declarations and class definitions

and that is included in both the main.cpp and CW4.cpp files. Functions/classes declared

in CW4.h will be visible/callable in any file that #include"CW4.h".

4. report.pdf file: the report.

If you have additional .cpp or .h files that you have written yourself and that are needed

for the completion of the assignment you may also submit these. However, do not submit

files that are created by the computer, including executables, object code, etc... You may

submit the .cbp file too, although I will be compiling and testing your code with standardized

functions that I write without it. In particular, one of the evaluation tests that will be carried

out will be changing the range of xMin, xStep, xMax, yMin, yStep, yMax and seeing if

your program can correctly recompute the MBS over a different range, so you should check

that your program works for different ranges.

During evaluation, a set of main.cpp template codes will be created that call the functions

and operators in your Complex class and check that each of your functions/operators are

producing correct outputs for the given inputs. Your own default main.cpp that you submit

should be setup to show how you tested and debugged your functions. An example main

could look as follows:1

in which I can easily see and uncomment/test/check each part of your submission with

appropriate testing functions written by you within each of the 3 main subfunctions. You

should also be aware of the fact that I will be subjecting your classes and functions to

standard tests that I will create for all students to check the correctness and completeness of

In addition, your programs will be tested for similarity with the programs of your classmates through a software to check for excessive duplicated code. Points will be deducted

from both parties for duplicate code, and in severe situations, the case may be raised for additional disciplinary actions. The exception will be for duplicated code that is given to you

in the practicals. But you are expected to comment your code to show your understanding

of the given code, and to differentiate yourself from your classmates.