% A few Matlab examples including 2D and 3D graphics, animations, and
% matrix operations.

set(0,'DefaultTextInterpreter','LaTeX')

%% 2d plot
clear, clf
x=-1:.1:1;
y=x.^3;
plot(x,y)
box off
xlabel('$x$')
ylabel('$y$')
title('$y=x^3$')

%% 3d plot
clear, clf
[x y]=meshgrid(linspace(-1,1,50));
z=x.^2+y.^2;
mesh(x,y,z)

%% 3d plot (improvement 1)
clear, clf
r=linspace(0,1,25);
vth=linspace(0,2*pi,25);
[r vth]=meshgrid(r,vth);
x=r.*cos(vth);
y=r.*sin(vth);
z=r.^2;
mesh(x,y,z) 

%% 3d plot (improvement 2)
clear, clf
axes('Pos',[.2 .1 .6 .8])
r=linspace(0,1,25);
vth=linspace(0,2*pi,25);
[r vth]=meshgrid(r,vth);
x=r.*cos(vth);
y=r.*sin(vth);
z=r.^2;
surfl(x,y,z+.5), hold on
shading interp, colormap copper
contour(x,y,z), hold off
set(gca,...
   'XLim',[-1 1],...
   'YLim',[-1 1],...
   'ZLim',[-inf inf],'ZTickL',[0:.5:1.5],'ZTickL',[-.5:.5:1],...
   'Box','on','Color',[1 1 1])
grid off
title('$z=x^2+y^2$')
xlabel('$x$')
ylabel('$y$')
zlabel('$z$')

%% movie
clear, clf
mov=avifile('example.avi')
x=linspace(-3,3,25);
y=linspace(-3,3,25);
[X Y]=meshgrid(x,y);
z=peaks(X,Y); 
surf(x,y,z)
shading interp
axis([-3 3 -3 3 -8 8])
set(gca,'Visible','off')
N=25;
for I=1:N
   surf(x,y,sin(2*pi*I/N)*z)
   shading interp
   axis([-3 3 -3 3 -8 8])
   set(gca,'Visible','off')
   mov=addframe(mov,getframe(gca));
end
mov=close(mov);
       
%% matrix multiplication
clear
N=100;
rand('state',1), A=ceil(5*rand(N));
rand('state',2), B=ceil(5*rand(N));
C=zeros(N);

tic
for I=1:N
   for J=1:N
      dmy=0;
      for K=1:N
         dmy=dmy+A(I,K)*B(K,J);
      end
      C(I,J)=dmy;
   end
end
toc

tic, C=A*B; toc

%% determinants, eigenvalues, etc.
clear
A=[1 0;0 5];
det(A)
[V D]=eig(A)
A*V
D*V

%% fourier series
clear, clf

t=linspace(-.5,.5,250);
f=.5*(square(t+.25)-square(t-.25));
h=[];
h=plot(t,f,'r','LineW',1); hold on

n=1:15;
A=[.5 sin(n*pi/2)./(n*pi/2)];
fn=A*cos(2*[0 n]'*pi*t);
h=[h plot(t,fn,'b','LineW',.5)]; hold off

set(gca,...
   'XLim',[-.5 .5],'XTick',[-.5 0 .5],...
   'YLim',[-.5 1.5],...
   'Box','off')
legend(h,{'$f(t)$',['$f_{' num2str(max(n)) '}(t)$']})
xlabel('$t$')

set(0,'DefaultTextInterpreter','TeX')