function exercise_1_6_4()
% 
%
% Written by:
% -- 
% John L. Weatherwax                2006-12-11
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all; 

m = 32;
A = delsq(numgrid('S',m)); 
issparse(A)
size(A)
fprintf( 'The size of delsq = (%d,%d)\n', size(A,1), size(A,2) ); 
fprintf( 'Number of non zeros of the matrix delsq = %d (=%f percent)\n', nnz(A), nnz(A)/prod(size(A)) ); 

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

pause; close all; 

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

pause; close all; 

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

pause; close all; 

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

pause; close all; 

% Part (e):
%
Anest = delsq(numgrid('N',m)); 
issparse(Anest)
size(Anest)
fprintf( 'The size of delsq(nested diss.) = (%d,%d)\n', size(Anest,1), size(Anest,2) ); 
fprintf( 'Number of non zeros of the matrix delsq(nested diss.) = %d (=%f percent)\n', nnz(Anest), nnz(Anest)/prod(size(Anest)) ); 

figure; spy(Anest); title( 'delsq(nested diss.)' ); 
tic, R = chol(Anest); t=toc; 
fprintf('Time to directly Cholesky factor delsq(nested diss.)= %f\n', t ); 
fprintf( 'Number of non zeros of the Cholesky factor of delsq(nested diss.) = %d (=%f percent)\n', nnz(R), nnz(R)/prod(size(Anest)) ); 
figure; spy(R); title( 'Cholesky factor of delsq(nested diss.)' ); 

return;