LAB 3, 02/12/14
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1.	A FOR-END loop is a simple control-flow feature for
	executing a set of commands a fixed number of times.
	
	For example:

		c = -0.1 ;

		z = 0  ;
		v = [z];

		for k = 1:1000
			z = z^2 + c ;
			v = [v ; z] ;
		end

	The first three statements are initializations.

	The counting variable k is internal to the loop; any
	symbol can be used.

	This simple program produces a column vector of 1001
	entries, with first entry equal to 0, and nth entry z_n
	given iteratively by

		z_n = (z_(n-1))^2 + c

	where c was the chosen value -0.1.


2.	A script file is a collection of MATLAB commands, saved
	with the .M extension:

		SCRIPT_FILE.M

	To execute these commands, make sure the file is in the
	working MATLAB directory, and simply type 

		SCRIPT_FILE

	in the command window.

	Your task: create the file 

		iterate_1.m

	consisting of the commands in 1. above with the exception
	of the very first line, i.e., the initialization of c.

	Then execute in the command window using

		c = -0.1 ;
		
		iterate_1


3.	The iteration of 2. above is the basis for the Mandelbrot
	fractal:

	http://www.math.utah.edu/~alfeld/math/mandelbrot/large.gif

	It consists of all points c on the complex plane for
	which the iteration in 1. (initialized at c) results in

		|z_n| less than or equal to 2 

	for ALL n.  For certain values of c, this property can
	be proved or disproved analytically; for other values, we
	have to rely on numerical results, which, of course, do
	not provide definite proof.

	Some examples (note: ABS(Z) is the modulus of Z):


		c = -0.1 ; 	iterate_1 ; plot(abs(v))

		c = -2 ; 	iterate_1 ; plot(abs(v))

		c = 1/4 ; 	iterate_1 ; plot(abs(v))

		c = j	; 	iterate_1 ; plot(abs(v))

		c = 1/2 ;	iterate_1 ; plot(abs(v))

		c = 2*j ; 	iterate_1 ; plot(abs(v))

	The first four values of c are inside the Mandelbrot set;
	the last two are outside. (Can be proved analytically.)


4.	As was seen above, the choices c=1/2 and c=2*j resulted
	in |z_n| exceeding 2 very soon, i.e., for small n.  To avoid
	needless iterations, we can use a WHILE loop instead
	of a FOR loop.  For example:
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
			
		z = 0  ;
		v = [z];

		while (abs(z)<=2) & (length(v)<=1000)

			z = z^2 + c ;
			v = [v ; z] ;

		end

	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	ensures that the loop is terminated as soon as abs(z)>2, with
	a maximum of 999 iterations.
	
	Note the use of the logical operator & in the WHILE condition
	above.  Logical and relational (e.g., <=) operators are 
	straightforward in MATLAB.  You can get documentation on them 
	using, e.g.,

		HELP & 

	(or other operator symbol, as appropriate).



5.	Save the code given above (between the two %%%%% lines) as

		iterate_2.m

	and try out the initial values

		c = -0.1, -2, 1/4, j, 1/2, 2*j

	with ITERATE_2 replacing ITERATE_1. In each case, determine

		max(abs(v))

	Your conclusions should be similar to 3. above.


6.	Next, we will use ITERATE_2 to test a large set of complex 
	numbers for inclusion in the Mandelbrot set. Specifically, we 
	consider all values c = x+j*y on the rectangular grid

		x = -2 : 0.04 : 0.5 ;

		y = -2 : 0.04 : 0.5 ;

	The c's that pass the test will be stored in a vector S.
	Since c=0 is known to be in the Mandelbrot set, we can null 
	out all c's that fail the test.  Thus for every complex value c
	tested, we will append

		c*( max(abs(v)) <= 2 )

	to S upon exiting the loop in ITERATE_2.

	The rectangular grid will be scanned using two nested FOR loops.

		S = [ ] ;

		for x = -2 : 0.04 : 0.5
		  for y = -2 : 0.04 : 0.5
		    c = x + j*y ;
		    iterate_2
		    S = [ S  c*(max(abs(v))<=2)]
		  end
		end

		plot(S, '+')