% define the matrix A
>> A = [.5,0; 0, 2]

A =

    0.5000         0
         0    2.0000

% define a ball in L2
>> [x] = L2_ball(.5);
>> size(x)

ans =

     2   501
% define the matrix A acting on each point in the ball  
>> for i=1:501, y(:,i) = A*x(:,i); end
>> subplot(2,2,1)
% plot the ball
>> plot(x(1,:),x(2,:))
>> hold on
>> subplot(2,2,2)
% plot the matrix A acting on the ball
>> plot(y(1,:),y(2,:))
>> hold on

% repeat
>> [x] = L2_ball(1);
>> subplot(2,2,1)
>> plot(x(1,:),x(2,:),'--')
>> for i=1:501, y(:,i) = A*x(:,i); end
>> subplot(2,2,2)
>> plot(y(1,:),y(2,:),'--')

% repeat
>> [x] = L2_ball(1.5);
>> for i=1:501, y(:,i) = A*x(:,i); end
>> subplot(2,2,1)
>> plot(x(1,:),x(2,:),'-.')
>> subplot(2,2,2)
>> plot(y(1,:),y(2,:),'-.')
>> axis([-3.2,3.2,-3.2,3.2])

% repeat with balls in L_inf
>> [x] = Linf_ball(.5);
>> size(x)

ans =

     2   401

>> subplot(2,2,3)
>> plot(x(1,:),x(2,:),'-')
>> hold on
>> for i=1:401, y(:,i) = A*x(:,i); end
>> subplot(2,2,4)
>> plot(y(1,:),y(2,:),'-')
>> hold on

% repeat
>> [x] = Linf_ball(1);
>> for i=1:401, y(:,i) = A*x(:,i); end
>> subplot(2,2,3)
>> hold on
>> plot(x(1,:),x(2,:),'--')
>> subplot(2,2,4)
>> plot(y(1,:),y(2,:),'--')


% repeat
>> [x] = Linf_ball(1.5);
>> for i=1:401, y(:,i) = A*x(:,i); end
>> subplot(2,2,3)
>> plot(x(1,:),x(2,:),'-.')
>> subplot(2,2,4)
>> plot(y(1,:),y(2,:),'-.')
>> axis([-3.2,3.2,-3.2,3.2])
>> title('A applied to balls in L_{inf}')

% put titles on
>> subplot(2,2,3)
>> title('Balls in L_{inf}')

>> subplot(2,2,2)
>> title('A applied to balls in L_{2}')

>> subplot(2,2,1)
>> title('Balls in L_{2}')

% print the figure
>> print -dps fig1.ps

% define a new matrix, and repeat the above
>> A = rand(2,2)

A =

    0.9501    0.6068
    0.2311    0.4860

>> eig(A)

ans =

    1.1586
    0.2775

>> figure(2)

>> [x] = L2_ball(.5);
>> for i=1:501, y(:,i) = A*x(:,i); end
>> subplot(2,2,1)
>> plot(x(1,:),x(2,:),'-')
>> hold on
>> subplot(2,2,2)
>> plot(y(1,:),y(2,:),'-')
>> hold on

>> subplot(2,2,1)
>> [x] = L2_ball(1);
>> for i=1:501, y(:,i) = A*x(:,i); end
>> plot(x(1,:),x(2,:),'--')
>> subplot(2,2,2)
>> plot(y(1,:),y(2,:),'--')


>> subplot(2,2,1)
>> [x] = L2_ball(1.5);
>> for i=1:501, y(:,i) = A*x(:,i); end
>> plot(x(1,:),x(2,:),'-.')
>> subplot(2,2,2)
>> plot(y(1,:),y(2,:),'-.')
>> title('A applied to balls in L_{2}')

>> subplot(2,2,1)
>> title('Balls in L_{2}')

>> subplot(2,2,3)
>> clear x y

>> [x] = Linf_ball(.5);
>> for i=1:401, y(:,i) = A*x(:,i); end
>> plot(x(1,:),x(2,:),'-')
>> subplot(2,2,4)
>> hold on
>> plot(y(1,:),y(2,:),'-')
>> subplot(2,2,3)
>> hold on

>> [x] = Linf_ball(1);
>> for i=1:401, y(:,i) = A*x(:,i); end
>> plot(x(1,:),x(2,:),'--')
>> subplot(2,2,4)
>> plot(y(1,:),y(2,:),'--')

>> subplot(2,2,3)
>> [x] = Linf_ball(1.5);
>> for i=1:401, y(:,i) = A*x(:,i); end
>> plot(x(1,:),x(2,:),'-.')
>> subplot(2,2,4)
>> plot(y(1,:),y(2,:),'-.')
>> title('A applied to balls in L_{inf}')

>> subplot(2,2,3)
>> title('Balls in L_{inf}')

>> subplot(2,2,4)
>> axis([-2.6,2.6,-2.6,2.6])

>> subplot(2,2,2)
>> axis([-2.6,2.6,-2.6,2.6])

>> print -dps fig2.ps

>> quit

 95559 flops.