%% Datta, page 595, Ex 25
% Demo_L2_3, check (insensitivities of the singular values
% Module 2, Lecture 2, Demo no. 1, SVD 
disp(' ----------- Singular Vale Decomosition: Demo no 3: -----------')
disp(' ---------------- (In)Sensitivities of the SVD ----------------')
disp('               ')
disp('Maya has to sow a slide again... ')
disp(' ')
wait


eeps=0.99; epss = 1e-5;
disp('eeps=0.99; epss = 1e-5;')

for n=[4,50]

%A=[1 1 1 1;0 eeps 1 1;0 0 eeps 1; 0 0 0 eeps];
%E=[0 0 0 0;0 0 0 0;0 0 0 0; epss 0 0 0];
A=ones(n);
A=triu(A);
dd=(1-eeps)*ones(n,1);
A=A-diag(dd);
A(1,1) = 1;
E = zeros(n); E(n,1) = epss;

AE = A+E;
[U,S,V] = svd(A);
[UE,SE,VE] = svd(AE);

EigA  = eig(A);
EigAE = eig(AE);
if n<8,
   disp(A),waitt('The matrix A')
   disp(E),waitt('The permutation matrix E')
   disp(A+E),waitt('The permuted matrix A+E')
   disp('The eigenvalues of A')
   format long
   disp(EigA)
   disp('The eigenvalues of A+E')
   disp(EigAE)
   disp('They are complex'),wait
   disp('_________________________________')
   disp('What about the sigular values?'),wait
   disp('Print those of A, of A+E and the difference')
   [diag(S) diag(SE) diag(S)-diag(SE)]
else
   figure(2),clf,mesh(A)
%   figure(3),clf,spy(A)
   waitt('.....')
   figure(2),clf,mesh(E)
   figure(4),clf, plot(EigA,'o'), title('The eigenvalues of A')
   figure(5),clf, plot(EigAE,'o'),title('The eigenvalues of A+E')
   figure(6),clf, plot(diag(S)-diag(SE),'o'),title('The difference in the singular values')
end
wait
end
disp('... end demo')




%