%
%  This script demonstrates the increasing relative
%  projections in the Least Sqares problem for a 
%  sequence of RHS b_j  of increasing norm that
%  all have the same projections onto Range(A)
%
%  To get a good aspect ratio on the plot
%  stretch the window by hand along the z - axis
%  after the first plot appears
%
%  D.C. Sorensen
%  4 Oct 01
%----------------------------------------------------
%

   for tau1 = [10 50 100 150],
   
      m = 200;
      t = linspace(0,(6.5*pi/4),m);
   
      x = cos(t);
      y = sin(t);
      z = t/10;
   
      x1 = [0 x(m)];
      x2 = [0 y(m)];
      x3 = [0 z(m)];
   
      z1 = [x(m) x(m)];
      z2 = [y(m) y(m)];
      z3 = 0*x3;
   
      plot3(x1,x2,z3,'b')
      hold
%
%     Plot four vectors of increasing length
%     that all have the same projection 
%     onto the Range(A)
%
   
      for tau = [10 50 100 150],
         plot3(x1,x2,tau*x3,'k')
         plot3(x1(2),x2(2),tau*z(m),'k+')
      end
   
      plot3(z1,z2,tau*x3,'m')
      
      t = linspace(0,2*pi,m);
      x0 = cos(t);
      y0 = sin(t);
      w = 0*t;
      mu = 2;
      x = x0*mu; 
      y = y0*mu; 
      plot3(x,y,w,'c');
      x = x0*mu; 
      
      tau = tau1 
%
%     Construct a perturbation to the RHS b
%
%     of size           mu*norm(b)  
%
%     where mu = .005  
%
   
      b = [x1(2)  x2(2) tau*z(m)];
      bnorm = norm(b);
      mu = .005*bnorm;
%
%     Plot the projection of the circle of such perturbations
%     in the Range(A) - in this case the x,y plane
%
      x = x1(2) + x0*mu; 
      y = x2(2) + y0*mu; 
      plot3(x,y,w,'r');
%
%     Plot the circle of such perturbations about b
%     parallel to the x,y plane
%
      w = tau*z(m)*ones(m,1);
      plot3(x,y,w,'k');
   
      grid
   
      hold
   if (tau1 < 20), 
      disp('  To get a good aspect ratio on the plot ')
      disp('  stretch the window by mouse along the z - axis ')
      disp('  after the first plot appears ')
   end
   disp('  ')
   disp('strike any key to continue')
   pause
   end