%
% emily_demo 
%
%   This demonstration of emily's abilities displays 5 precomputed matrices:
%
%    1. The FFT of the 100x100 identity matrix: fft(eye(100))
%    2. The FFT of the 324x324 5 pt. finite difference Lapacian on a square
%       domain: fft(full(delsq(numgrid('S',20))))
%    3. The inverse of the 196x196 5 pt. finite difference Lapacian on a square
%       domain: inv(delsq(numgrid('S',16)))
%    
%       The last two matrices are accumulations of the residual (R) and 
%    search (D) vectors during a run of the CG (Congugate Gradient) algorithm 
%    on the test matrix A.  A was generated as follows:
%
%         t2 = 1.0/(n*n);
%         for k=1:n
%           A(k) = k*k*t2;
%         end;
%         A = diag(A);
% 
%    The final two matrices are:
%
%    4. R transpose times R after 100 CG iterations on A above.
%    5. D transpose times A times D after 100 CG iterations on A above.
%

load demo
disp('fft of the 100x100 identity matrix (try changing the scaling to linear)');
emily(fft100);
disp('324x324 matrix of the FFT of the 5 pt. Laplacian on a square domain');
emily(fftinv324);
disp('196x196 matrix of the inverse of the 5 pt. finite difference Laplacian on a square domain');
emily(Lap196);
disp('R''*R after 100 CG iterations on a test matrix');
emily(RtR);
disp('D''*A*D after 100 CG iterations on a test matrix');
emily(DAD);