1. Computation and Simulation EE317 2008-2009 Assignment One
Group 6 :
Student Name: Student Number
Cong Chen
Shane Conaty
John Maguire

2. WHAT IS A FRACTAL ?
A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity.
The word ‘fractal’ was derived from the Latin ‘fractus’ meaning "broken" or "fractured by Benoît Mandelbrot in 1975.
A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.
The Mandelbrot set is a famous example of a fractal.

3. OCCURRENCE AND USE OF FRACTAL IN LIFE
First the study of ‘fractal’ has create a generation of various arts form.
Images of fractals can be created using fractal generating software.
‘Fractal Art contest’ has taken place every years.
Fractal music is also popular world wide.

4. OCCURRENCE AND USE OF FRACTAL IN GAMING
Fractals are often used in Gaming industry for making backgrounds and creating CG animations
One very famous example is the CG animation in series of ‘Final Fantasy’

5. OCCURRENCE AND USE OF FRACTAL IN ENGINEERING
Fractals are often used in ‘Fracture Mechanic’.
Fracture mechanics is the field of mechanics concerned with the study of the formation of cracks in materials.
It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.

6. OCCURRENCE AND USE OF FRACTAL IN ENGINEERING II
Fractal has also been used in ‘Rock and Soil Mechanic’
Fractal dimension will be used to characterize the state of rolling bearing failure in performance under different non-linear behavior.
The fractal dimension can be used as a means of identification of the features of rolling bearing failure.
Hydropower projects with many rocks, soil and concrete materials in close contact, and soil and rocks exist substantial distribution of irregular fissures or pores

7. OCCURRENCE AND USE OF NEWTONRAPHSON’S METHOD IN ENGINEERING
In numerical analysis, NewtonRaphson method (named after Isaac Newton and Joseph Raphson) is perhaps the best known method for finding successively better approximations to the zeroes (or roots) of a real-valued function. Newton's method can often converge remarkably quickly, especially if the iteration begins "sufficiently near" the desired root.

8. OCCURRENCE AND USE OF NEWTONRAPHSON’S METHOD IN ENGINEERING
Newton-Raphson’s method is used in the application of coupling chaos mapping Newton iterative method in MOTOR SYSTEM.
Newton-Raphson’s method can also be used to get more accurate approximations where needed. 
For example Rocket trajectory for space shuttles needs to be calculated to perfection because a minor miscalculation here on earth could mean thousands of miles in the difference from predicted position and actual position in space.

9. DESIGN AND CREATE FRACTAL IMAGE BY USING MATLAB
Y=X^3 - 1

10. CODING close all; clear all;
real_lower = -5; // create a plane that has value -5
real_upper = 5; // to 5 on x-axis, and -5 to 5 on the
imag_lower = -5; // y- axis
imag_upper = 5;
N =1000; // initial all the values
step_1 = (real_upper - real_lower)/N ;
step_2 = 1;
Max = 20 ; // the newton-raphson’s method will
threshold = 1e-12; // run 20 time, and it has a threshold //value of 10 to the power of -12
root_values = zeros(N) ;

11. CODING % define 3 roots root1 = 1.0 ; root2 = -0.5 - i*sqrt(3.0)/2.0 ;
The function p(z) = z3 − 1) has three roots, one real root and two complex roots.
There are:
1.0 ;
i*sqrt(3.0)/2.0 ;
i*sqrt(3.0)/2.0 ;
We need to define them first.

12. CODING
// because of we set N equal to //1000, that means we put //1,000,000 points on the plane, //the Newton-raphson’s //method will run through all //these 1,000,000 points to find //corresponding roots
// set the initial step equal to 1, //and maximum step as 20. //That means Newton-//raphson’s method will run 20 //times for each points in order //to find the accurate roots.
// y is the function on it’s own
// z is the first derivative of the // original function
// x_new = x_old - y/z is the // Newton-raphson’s method
for(ct1 = 1:N)
for(ct2 = 1:N)
x_old = real_lower + ct2*step_1 + i*(imag_lower + ct1*step_1) ;
fprintf(1,'x_old equals %f %f \n',real(x_old),imag(x_old)) ;
step_2 = 1 ;
while (step_2 < Max)
%fprintf(1,' x guess equals %1.14f \n',x_old);
y = power (x_old,3) - 1;
z = 3.0 * power(x_old,2) ;
x_new = x_old - y/z;
x_old = x_new ;
step_2 = step_2 + 1;
end

13. CODING
if( abs(x_new - root1 ) < threshold ) // if the roots we get is
root_values(ct1,ct2) = 0.5; //close to root1, then we
end //set it as 0.5.
if( abs(x_new - root2 ) < threshold ) // if the roots we get is
root_values(ct1,ct2) = 1.5; // close to root1, then we
end // set it as 1.5
if( abs(x_new - root3 ) < threshold ) // if the roots we get is
root_values(ct1,ct2) = 2.5; // close to root1, then we
end // set it as 2.5
end
end //end the loop

14. CODING mymap = [ 1 0 0 ; 0 1 0 ; 0 0 1] ; // set up my own
//color map.
Figure
surf(root_values) // the final step is to
view([0 90]) // draw the image
shading interp // on the screen.
colormap(mymap)
colorbar

15. THE OTHER TWO IMAGE THAT REQUIRED
This is the fractal image of function Y=X^4 – 1
This is the fractal image of function Y=X^3-2X+2

16. Discuss a specific innovation that you have developed in completing this assignment.
During the assignment, we can found the program that we wrote is not very efficient, because of the method we used was to set up a max value of iteration, 20. The Newton-Raphson’s method will run 20 of time to find the roots. It took a very long time.
For improvement, we can put decision loop in to the Newton-Raphson method loop, as soon as the method find the root, then we can decide which root it belongs to, then jump out the loop or trying to find another root.