%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  Solution to Homework 3, problem 3.
%  Solve the linear system Au=b where A is the
%  finite difference approximation to 
%       - u_{xx} - u_{yy} -u_{zz} = f(x,y,z)
%  for (x,y,z) in (0,1) x (0,1) x (0,1), 
%  with u = 0 on the boundary of the box.
%
%  Use varying meshes,  with h=1/(n+1), and solve using
%  preconditioned CG, with the preconditioner an incomplete
%  Cholesky factorization with varying drop tolerance.
%
%  Plot the time and memory required by the various 
%  solution algorithms.
% 
%  problem3.m    Dianne O'Leary  04-2005
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


j = 0;

% Initialize the colors for plotting.

cc = ['b','g','r','c','m','k','b','g','r','c','m'];

% Loop for n=10:5:30
%  Generate the problem
%  Solve for various values of droptol 

for n=10:5:30
   j = j + 1;
   i = 0;
   [A,b,h] = laplace3d(n);
   tol = .01*h^2;
   normb = norm(b);
   if (n < 30)
      tryd = [0 .0001 .001 .01 .05 .1 .2 .3 ];
   else
      tryd = [.001 .01 .05 .1 .2 .3 ];  % otherwise: memory troubles
   end
 
   for droptol = tryd

     i = i + 1;
     disp('  ')
     disp(sprintf('Solving with n=%d and convergence tolerance = %5.2e', ...
                length(b),tol))
   
      tic
      R = cholinc(A,droptol);
      [x,cflag(i,j),relres,iter(i,j)] = pcg(A,b,tol,500,R',R);
      nz(i,j) = 3*nnz(A) + 5*n + 3*nnz(R);
      ctime(i,j) = toc;
      rnorm(i,j) = norm(b-A*x)/normb;
   end
end

% Plot the results.

hold off
for jj=1:j,
  loglog(nz(:,jj),ctime(:,jj),cc(jj))
  hold on
  loglog(nz(:,jj),ctime(:,jj),'*')
end

xlabel('memory')
ylabel('time')

print -depsc prob3.eps