LAB 6, 03/05/2014
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1.	The functions 

		MIN, MAX, SUM, PROD, SORT (ascending order)

	have obvious meaning when applied to numerical (row or column) 
	vectors.  When applied to arrrays, the operations are performed
	separately on each column.  For example,

		A

		min(A)

		min(min(A))

		sum(A)

		sum(sum(A))

		(prod(A'))'		% row, instead of column, products

		sort(A)

		-sort(-A)		% columns sorted in descending order



2.	So far we have considered numeric and logical arrays.  Another
	type of array is the so-called string array, which consists of
	ASCII characters.

	String arrays are specified using quotes, which may enclose
	single or multiple characters.  The quotes also help distinguish
	string arrays from variable names.

	Examples:

		A			% variable name
					% values displayed with indent					

		'A'			% string 
					% displayed without indent

		string1 = 'A$>'

		size(string1)

		string2 = ['A' '$>']

		string1 == string2



	Two-dimensional string arrays can be also specified (e.g., to
	display multiple lines of text), as long as all rows have the
	same number of elements:


		['nineteen characters' ; 'followed by twenty-two']

		['insert spaces' ; '  as needed! ']
	
	
	In time, we will examine some important links (and functions that 
	enable them) between strings and numerical arrays.


3.	The NxN system of linear equations

		A*x = b

	is solved using the backslash operator

		x = A\b

	Matlab will adaptively choose a specific algorithm to solve the 
    	systems of linear equations according to the structure of the matrix A.
	For more information, see the documentation for MLDIVIDE.

	Notationally, this is equivalent to

		x = inv(A)*b

	The latter syntax is correct, but it (generally) involves more 
    	operations than A\b and possibly more round-off errors. To 
	compare the error in the two computations, consider

		for i = 1:1000			% number of trials
			A = rand(100,100) ;	% random entries in (0,1)		
			x = rand(100,1) ;	% the solution
			b = A*x ;

			xsol1 = A\b ;
			xsol2 = inv(A)*b ;

			e(i) = ( sum( abs(x-xsol1) ) < sum( abs(x-xsol2) ) );
		end

		sum(e)/1000         % frequency with which
                            % A\b has lower abs. error

	A novel feature in the FOR loop above was the on-the-fly creation
	of the (logical) vector e, indexed by the loop variable i.  There
	was no need to initialize e.


4.	Of course, A*x = b has no solution if the matrix A is singular
	(has no inverse).  For example:

		A = [ 3 1 -2 11 ; 1 8 2 -3 ; 4 15 7 1 ; 1 2 -5 4 ] 

		b = A*[1 1 1 1].'

		A\b

	Singularity can be confirmed by computing the determinant (DET), 
	which is nonzero if and only if the matrix is nonsingular.  In this 
	case,

		det(A)

	produces a very small value (only a couple of orders of magnitude
	higher than the machine epsilon, 2^(-52)) instead of the correct
	answer (0).  This is (presumably) because MATLAB obtains the
	determinant through Gaussian elimination, as the product of all
	pivot elements (with a sign change if an odd number of row
	interchanges were performed).

	An alternative (and standard) expression for the determinant of a
	NxN matrix is the sum of N! products, each product containing N 
	terms of the matrix A.  If the matrix has integer entries, then the 
	determinant is also an integer; thus a small answer such as the one
	obtained above can be rounded off to 0.

	The rank of a matrix (obtained by the function RANK) is the largest 
	number of linearly independent columns - or equivalently, rows - that 
	can be drawn from that matrix.  A NxN matrix is nonsingular if and
	only if it has rank equal to N.  Compare


		rank(A)             % singular, as we saw earlier

		rank(rand(4,4))     % almost certainly nonsingular

		rank(rand(4,5))     % why?

		rank(rand(5,4))     % why?
	

5.	In the last example, we test all 3x3 matrices whose entries take 
	values 5, 6, 7 or 8 for singularity.   We do so using a single FOR 
	loop, with loop index i = 0:4^9-1.  The index is converted to a 
	base-4 integer, which is then read into the columns of a 3x3 matrix A, 
	after incrementing each digit by 5.  Three new functions are used:

		DEC2BASE(D,B,K)   converts a decimal integer D into a B-ary
                          string with a total of K digits

		STR2NUM(S) 	converts numeric characters to actual integers

		BLANKS(K)	string of K blanks (row vector)

	Also, the syntax

		A(:)        concatenation of all columns of A into a 
                    single column vector

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

		c = 0 ;                          % initialize # singular matrices
		B = blanks(9) ;

		for i = 0:4^9-1
			S = dec2base(i,4,9) ;
			S = [S;B] ;
			S = S(:) ;                   % 4-ary digits separated by spaces
			A = str2num(S) + 5 ;
			A = reshape(A,3,3) ;
			c = c + (abs(det(A))<0.5) ;  % compensate for round-off
		end
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

	Thus 25,252 out of a total of 4^9 = 262,144 such matrices are singular.