%
% 12.747 Lecture 10; Demo 1
% 
% Calculation of a simple under-determined problem using SVD
% S. Doney, Sept. 30th 2002

% Design model and compute data for a 
% under-determined system with 2 data points
% and 3 model parameters
n=2;
m=3;
r=2;
%
disp('n m r')
[n m r]
pause
disp('design matrix')
A=[1 0 0; 0 1/2 1/2]
pause
x_true = [2 1 -1]';
b_true = A*x_true;
% compute SVD and examine the singular values 
[U,S,V] = svd(A);
disp('compute singular value decomposition')
S
pause
% only keep the first r non-zero eigenvalues
disp('only keep 2 non-zero eigenvalues')
S_r = S(:,1:r)
pause
W_r = diag(1./diag(S_r));
% only keep the first r columns of V
U_r=U;
U_r
pause
V
pause
V_r=V(:,1:r);
V_r
pause
x_est = (V_r*W_r*U_r')*b_true;
disp('x_est = (V_r*W_r*U_r^T)*b_true')
disp('x_true x_est')
[x_true x_est]
pause
% check solution
disp('check solution b_true A*x_est')
[b_true A*x_est]
pause
% check null space 
disp('check null space assumption A*V(:,3)')
A*V(:,3)
pause
%compute model parameter resolution
disp('compute model resolution V_r*V_r^T')
V_r*V_r'
pause
%compute data resolution
disp('compute data resolution: U_r*U_r^T')
U_r*U_r'
pause