% Datta, page 594, Ex 22
% verification of a bound of the smallest singular value
% of a bidiagonal matrix
% Module 2, Lecture 2, Demo no. 1, SVD 
disp(' ---------- Singular Vale Decomosition: Demo no 2: --------------')
disp(' ---------- Theoretical estimates for singular values -----------')
disp('               ')
disp('Maya has to show a slide first... ')
wait

disp('Beta = [5 100 10000 1e+6];')
disp('Eta  = [0.5 1e-2 1e-4 1e-6];')
disp('  ')
disp('  ')
wait
Beta=[5 100 10000 1e+6];
Eta  =[0.5 1e-2 1e-4 1e-6];

for k=1:length(Beta),
beta=Beta(k);
eta  =Eta(k);
n=20;
A=bidiagonal_estim(beta,eta,n);
[U,S,V] = svd(A);

dS=diag(S);
% if k==1, 
%    figure(1),clf,plot(dS)
%    title('figure(1),clf,plot(dS)')
%    waitt('Not seen well.')
% end
figure(2),clf,semilogy(dS)
title('figure(2),clf,semilogy(dS)')
waitt('bla')

sigma_min = min(dS);

estimate=beta^(1-n)*(1-(2*n-1)*eta);
disp(['Parameters: beta=' num2str(beta) ',  eta=' num2str(eta)])
disp(['Computed value ' num2str(sigma_min)])
disp(['Estimated value ' num2str(estimate)])
wait
end

disp('We can see one portrait of the matrix:  ')
wait
figure(3),clf,mesh(A)
disp(' ... end demo.')
%