%%
clear
format compact

%%
% The functions sin^2(t), cos^2(t), and 1 are linearly dependent; 
% the following matrix is ill-conditioned.
beta = 1 + 4.5e-8;
t = linspace(0, 3, 400)';
A = [ sin(t).^2, cos(beta*t).^2, t.^0 ];
kappa = cond(A)

%%
% Now we set up an artificial linear least squares problem with a known
% exact solution that actually makes the residual zero.
x = [1; 2; 1];
b = A*x;  

%%
% If we formulate and solve via the normal equations, we get a large
% relative error. 
N = A'*A;
x_NE = N \ (A'*b);
observed_err = norm(x_NE - x)/norm(x)