%function exercise_1_6_6()
%
% Written by:
% -- 
% John L. Weatherwax                2006-12-30
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all; 

nprime = 20; mprime = 15; 
A = gallery( 'wathen', nprime, mprime ); 
issparse(A)
size(A)
fprintf( 'The size of wathen(%d,%d), = (%d,%d)\n', nprime, mprime, size(A,1), size(A,2) ); 
fprintf( 'Number of non zeros of the matrix wathen = %d (=%f percent)\n', nnz(A), nnz(A)/prod(size(A)) ); 

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

pause; close all; 

% Part (b):
%
p = randperm(size(A,1));
arnd = A(p,p); 
figure; spy(arnd); title( 'Randomized version of wathen' ); 
tic, R = chol(arnd); t=toc;
fprintf('Time to factor randomly Cholesky factor wathen = %f\n', t ); 
fprintf( 'Number of non zeros of the Cholesky factor of wathen = %d (=%f percent)\n', nnz(R), nnz(R)/prod(size(A)) ); 
figure; spy(R); title( 'Cholesky factor of randomized wathen' ); 

pause; close all; 

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

pause; close all; 

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

return;