% function norm_demo(A)
% 
% Illustrates matrix 1-, 2-, and infinity-norm for 2-by-2 matrices.
% After Trefethen and Bau, Figure 3.1.

 function norm_demo(A)

 if (max(size(A)) ~= 2) | (min(size(A)) ~= 2) | ~(isreal(A))
    fprintf('A must be a real 2-by-2 matrix\n')
    return
 end
 
% compute scale factor for the plot axes
 ax = 1.05*max([1 norm(A,1) norm(A,2) norm(A,inf)]);    

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1-NORM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% sample vectors of the form [x(1,k);x(2,k)] that are on the 1-norm unit ball,
% i.e., |x(1,k)|+|x(2,k)| = 1.

 x1 = linspace(0,1,100);
 x2 = [1-x1 (fliplr(x1)-1)]; x2 = [x2 fliplr(x2)];
 x1 = [x1  fliplr(x1)]; x1 = [x1 -fliplr(x1)];
 x = [x1 -fliplr(x1); x2 fliplr(x2)];

% plot the 1-norm unit ball

 figure(1), clf
 subplot(1,2,1)
 plot(x(1,:),x(2,:),'b-','linewidth',2), axis equal, axis(ax*[-1 1 -1 1])
 title('1-norm unit ball') 
 hold on

% find the vector on the unit ball that maximizes A*x.
% for the 1-norm, this is +/- [1;0] or +/- [0;1].
% plot it as a red line.

 if (abs(A(1,1))+abs(A(2,1))) > (abs(A(1,2))+abs(A(2,2)))
    xmax = [1;0];
 else
    xmax = [0;1];
 end
 ymax = A*xmax;
 plot([0 xmax(1)], [0 xmax(2)], 'r-','linewidth',2) 
 
% now show the effect of applying A to these vectors on the unit ball.
 
 y = A*x;
 subplot(1,2,2)
 plot(y(1,:),y(2,:),'b-','linewidth',2), axis equal, axis(ax*[-1 1 -1 1])
 title('A*(1-norm unit ball)')
 hold on

% plot the maximal A*x in red (computed above)
 plot([0 ymax(1)], [0 ymax(2)], 'r-','linewidth',2) 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 2-NORM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% sample vectors of the form [x(1,k);x(2,k)] that are on the 2-norm unit ball,
% i.e., |x(1,k)|^2+|x(2,k)|^2 = 1.

 theta = linspace(0,2*pi,400);
 x = [cos(theta); sin(theta)];

% plot the 2-norm unit ball

 figure(2), clf
 subplot(1,2,1)
 plot(x(1,:),x(2,:),'b-','linewidth',2), axis equal, axis(ax*[-1 1 -1 1])
 title('2-norm unit ball') 
 hold on

% find the vector on the unit ball that maximizes A*x.
% for the 2-norm, we can compute this using the 
% singular value decomposition, covered in a later lecture.
% plot this vector as a red line.

 [U,S,V] = svd(A);
 xmax = V(:,1);
 ymax = A*xmax;
 plot([0 xmax(1)], [0 xmax(2)], 'r-','linewidth',2) 
 
% now show the effect of applying A to these vectors on the unit ball.
 
 y = A*x;
 subplot(1,2,2)
 plot(y(1,:),y(2,:),'b-','linewidth',2), axis equal, axis(ax*[-1 1 -1 1])
 title('A*(2-norm unit ball)')
 hold on

% plot the maximal A*x in red (computed above)
 plot([0 ymax(1)], [0 ymax(2)], 'r-','linewidth',2) 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% INFINITY-NORM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% sample vectors of the form [x(1,k);x(2,k)] that are on the infinity-norm unit ball,
% i.e., max{|x(1,k)|,|x(2,k)|} = 1.

 x = linspace(-1,1,100);
 o = ones(1,100);
 x = [x o fliplr(x) -o; 
      o fliplr(x) -o x];

% plot the infinity-norm unit ball

 figure(3), clf
 subplot(1,2,1)
 plot(x(1,:),x(2,:),'b-','linewidth',2), axis equal, axis(ax*[-1 1 -1 1])
 title('\infty-norm unit ball') 
 hold on

% find the vector on the unit ball that maximizes A*x.
% for the infinity-norm, this is +/- [1;+/- 1].
% plot this vector as a red line.

 if norm(A*[1;1],inf)>norm(A*[1;-1],inf)
    xmax = [1;1];
 else
    xmax = [-1;1];
 end
 ymax = A*xmax;
 plot([0 xmax(1)], [0 xmax(2)], 'r-','linewidth',2) 
 
% now show the effect of applying A to these vectors on the unit ball.
 
 y = A*x;
 subplot(1,2,2)
 plot(y(1,:),y(2,:),'b-','linewidth',2), axis equal, axis(ax*[-1 1 -1 1])
 title('A*(\infty-norm unit ball)')
 hold on

% plot the maximal A*x in red (computed above)
 plot([0 ymax(1)], [0 ymax(2)], 'r-','linewidth',2)