% m11_leastsquares3.m - least-squares by three methods

  t = 2*pi*(1:1000)'/1000;                                  
  b = t/5 + (t<pi);                                       
  plot(t,b), grid on                                      
  A = [ cos(0*t) sin(t) cos(t) sin(2*t) cos(2*t) ];       
                                                          
% Backslash (which uses QR factorisation):                
  x1 = A\b                                                
  hold on                                                 
  plot(t,A*x1)                                        
  norm(A*x1-b)                                            
                                                          
% Explicit use of QR factorisation:                       
  [Q,R] = qr(A,0);                                        
  x2 = R\(Q'*b);                                          
  norm(x1-x2)                                             
                                                          
% Normal equations:                                       
  x3 = (A'*A)\(A'*b)                                      
  norm(x3-x1)