%% HW2 - 3101 Matlab -- (INSERT NAME and UNI HERE)
% Due 11:59pm on Tuesday, March 25th, 2008
%% Goals
% Start using matlab as a simple scripting language

%% Directions
% Please fill in your name at the top of this page and write your answers
% under their corresponding question.  When you are done, publish this file
% to html using File > Publish To HTML (or the command publish('hw1.m', 'html')
% and create a new zip file that is labeled with your UNI and homework 
% number (eg. bs2018-hw1.zip), that contains this original m file, and the 
% html directory created from publishing the file. Email this file to
% cs3101@gmail.com, making sure to include your name, and uni in the subject.


%% Note
% When I say display a variable I mean display it so the output is labeled
% with the variable name, don't use disp, just leave off the semicolon
% from the end of the command. 

%% Help
% If you get stuck, type the command doc to pull up the documentation
% window, from there it's easy to search for any other commands or
% functions you might want to use...


%% Problem 1 - Fibonacci numbers and the golden ratio
% (http://en.wikipedia.org/wiki/Fibonacci_number) Use a for loop to compute
% the first 100 Fibonacci numbers, and store them in a variable called f.
% only display the first 10 elements, make sure not to display anything
% while you are looping (use a semicolon)


%% 1.2
% plot the first 20 Fibonacci numbers, and title your plot.


%% 1.3
% (http://en.wikipedia.org/wiki/Golden_ratio) The golden ratio (also known as
% phi) is an irrational number that can be found throughout science, art
% and architecture since the renaissance... we know that phi is equal to (1
% + sqrt(5)) / 2, however it is also easily expressed as the ratio between
% two successive Fibonacci numbers (weird... i know...). use a while loop
% to find the first two fibonacci numbers whose ratio approximates phi with
% an error less then 0.001, and print out only those two successive
% Fibonacci numbers... note: the while loop should terminate after it has
% found the first set of Fibonacci numbers whose ratio yields an error less
% then 0.001


%% 1.5
% display the exact phi, then display your approximation for phi calculated
% above, and then display the absolute value of their difference



%% 1.5
% change the display format to long and then display the exact phi,  the
% approximate phi, and their difference



%% 1.6
% change the display format back to short (the default)



%% Problem 2 - Exploring the julia set fractals
% (http://en.wikipedia.org/wiki/Julia_set) Julia set fractals are very
% interesting, and beautiful; they are also not too hard to create in
% matlab.  first we start with a grid of equally spaced complex numbers,
% where the real components vary from -2 to 2, and the imaginary components
% vary from -2i to 2i.  then for each number in this grid, we recursively
% apply a mathematical function to it many times and see how many times
% before the complex number becomes bigger then some cutoff value... that
% count of how many iterations it takes to blow up determines the color of
% that one pixel of the fractal


%% 2.1
% first lets start with a matrix of all zeros, so let's set N equal to 10,
% and make a new matrix F which is NxN and contains all zeros



%% 2.2
% create a variable called x which contains N numbers that are equally
% spaced between -2 and 2, display x


%% 2.2.1
% how many elements are in x?


%% 2.3
% create a variable called z0 which is a matrix of size NxN, such that
% element i,j of z0 should equal x(i) + sqrt(-1)*x(j).  When we are done
% filling z0, it should contain equally spaced complex numbers such that
% the real component goes from -2 to 2 horizontally, and imaginary
% component goes from -2i to 2i vertically. Display z0. (Extra credit, if
% you can make z0 using only 1 line, no loops or semicolons)


%% 2.4
% for each element i,j in F, get the initial value of z from the matrix z0
% and then use a while loop and a counter variable to count how many
% iterations of applying the function z = z^2 + c causes the absolute value
% of z to become bigger then the variable blowup.  After you are done,
% F(i,j) should equal the number of times you had to apply z = z^2 + c
% repeatedly, before abs(z) gets bigger then the blowup parameter. Note:
% make sure to put a limit on the number of times the inner while loop can
% iterate if abs(z) doesn't blowup.

c=0.285+0.01*sqrt(-1)
blowup = 200;
maxiter = 200;


%% 2.5
% display F, and then plot it using imagesc.  make sure the axes are set to
% image



%% 2.6
% Collect all the code from this problem, and put it together as 1 chunk of
% code below and set N to be much bigger, like between 100 and 500



%% 2.7
% Make your own fractal! you can change the value for c, or pick a different function
% instead of z = z^2 + c, also try picking a colormap that is different
% then the default one... pick a setup that you find interesting and
% calculate it and plot it below