%Given a ill-posed problem: Af=g where A has a very large condition number

%a) singular values

sigma=svd(A);

figure;

plot(sigma,'b-','linewidth',1.5);

title('Singular values');

%b) condition number

k=cond(A,2);

disp('The condition number of A is:');

disp(k);

%c) pseudosolution

%fs=sum(Vk*Uk'*g/sigma)

[u,s,v]=svd(A);

sigma=diag(s);

fs=zeros(size(g));

for ii=1:rank(s);

   temp=(1/sigma(ii))*v(:,ii)*u(:,ii)'*g;

   fs=fs+temp;

end

figure;

plot(fs,'b-','linewidth',1.5);

title('Pseudosolution');

%Alternatively

%r=rank(A);

%sinv=diag(1./sigma);

%fs=v(:,1:r)*sinv(1:r,1:r)*u(:,1:r)'*g;

% discrepancy

%epsillon=sqrt(Af-g);

eps=abs(A*fs-g);

figure;

plot(eps,'b-','linewidth',1.5);

title('Discrepancy');

% energy E

E=fs.^2;

figure;

plot(E,'b-','linewidth',1.5);

title('Energy');

%Note the problem can also be solved by QR factorization or householder reflectors. But SVD is more preferable inasmuch it is robust to noise, able to hand rank-deficiency and efficient in computation as well.