function exercise_1_6_3()
% 
%
% Written by:
% -- 
% John L. Weatherwax                2006-03-28
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all; 

load west0479;
A = west0479;
fprintf( 'The size of west0479 = (%d,%d)\n', size(A,1), size(A,2) ); 
fprintf( 'Number of non zeros of the matrix west0479 = %d\n', nnz(A) ); 
%issparse(A)
figure; spy(A); title( 'west0479' ); 
A = A' * A;
figure; spy(A); title( 'west0479'' * west0479' ); 
%issparse(A)
fprintf( 'Number of non zeros of the matrix west0479'' * west0479 = %d\n', nnz(A) ); 

% Part (a):
%
tic, R = chol(A); t=toc; 
fprintf('Time to directly Cholesky factor west0479'' * west0479 = %f\n', t ); 
fprintf( 'Number of non zeros of the Cholesky factor of west0470'' * west0470 = %d\n', nnz(R) ); 
figure; spy(R); title( 'Cholesky factor of west0479'' * west0479' ); 

% Part (b):
%
p = randperm(479);
arnd = A(p,p); 
figure; spy(arnd); title( 'Randomized version of west0479'' * west0479' ); 
tic, R = chol(arnd); t=toc;
fprintf('Time to factor randomly Cholesky factor west0479'' * west0479 = %f\n', t ); 
fprintf( 'Number of non zeros of the Cholesky factor of west0470'' * west0470 = %d\n', nnz(R) ); 
figure; spy(R); title( 'Cholesky factor of randomized west0479'' * west0479' ); 

% Part (c):
%
p = symrcm(A);
arcm = A(p,p); 
figure; spy(arcm); title( 'Reverse Cuthill-McKee ordered version of west0479'' * west0479' ); 
tic, R = chol(arcm); t=toc;
fprintf('Time to factor the reverse Cuthill-McKee ordered Cholesky factor west0479'' * west0479 = %f\n', t ); 
fprintf( 'Number of non zeros of the Cholesky factor of west0470'' * west0470 = %d\n', nnz(R) ); 
figure; spy(R); title( 'Cholesky factor of reverse cuthill-mckey ordered west0479'' * west0479' ); 

% Part (d):
%
p = symmmd(A);
ammd = A(p,p); 
figure; spy(ammd); title( 'Minimum degree ordering of west0479'' * west0479' ); 
tic, R = chol(ammd); t=toc;
fprintf('Time to factor the minimum degree ordering Cholesky factor west0479'' * west0479 = %f\n', t ); 
fprintf( 'Number of non zeros of the Cholesky factor of west0470'' * west0470 = %d\n', nnz(R) ); 
figure; spy(R); title( 'Cholesky factor of the minimum degree ordering of west0479'' * west0479' );