%% Datta, page 595-596, Ex 4 + test data
% Compute the rank of a matrix
% Module 2, Lecture 2, Demo no. 4, SVD 
disp(' ----------- Singular Vale Decomosition: Demo no 4:  -------------')
disp(' ----------------------- numerical rank --------------------------')
disp('')
waitt('Create the Kahan marix')
c=0.2; s=0.6; n=100;
A=Kahan(s,c,n);
disp('Compute the singular value decomposition of A')
tic,[U,S,V]=svd(A);
disp(['It takes ' num2str(toc) ' sec.'])

figure(1),clf,plot(diag(S))
figure(1),clf,semilogy(diag(S),'o-')
title('figure(1),clf,semilogy(diag(S))')
wait

disp([' Check rank(A):' int2str(rank(A))])
disp(' Check sd=diag(S); sd(61:65)')
sd=diag(S);
sd(61:65),wait
disp('Check low rank approximations:')
%
%% Test low rank approximations of the matrix
for k=[20,50,61,62,65,80]
    AK=U(:,1:k)*S(1:k,1:k)*V(:,1:k)';
    nrm=norm(A-AK);
    disp(['k= ' int2str(k) ', norm(A-AK) ' num2str(nrm) ', sigma(k+1) ' num2str(S(k+1,k+1))])
end
disp('...end demo.')