%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% function gd2(x0) - minimizes f(x)=cos(x)*cos(y)
%%%
%%% INPUT: x0 = 2 by 1 column vector for starting point s.t. -4<x,y<4
%%% EXAMPLE; gd([1;2])
%%% Run this for different starting points
%%%
%%% SAMSI NDHS Undegraduate Workshop May 2006
%%% created: Anjela Govan, North Carolina State University
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
function gd2(x0)
hold off
clf

%% points for the surface
X0= [-4: .1: 4];
Y0= [-4: .1: 4];

%% Create mesh plot with contour plot of f(x)=cos(x)*cos(y)
[X,Y] = meshgrid(X0,Y0);
Z=cos(X).*cos(Y);
figure(1)
surfc(X,Y,Z)    % shows 3D plot of surface f(x)=cos(x)*cos(y)
shading('faceted')

%%Contour graph of the surface of f(x) and plot the initial point
v= cos(X0).*cos(Y0);
figure(2)
[C,H]=contour(X,Y,Z);
hold on
plot(x0(1),x0(2),'xk')


xnew=x0;              % user-defined starting point
gradient = inline('[sin(x)*cos(y), cos(x)*sin(y)]');
gradnew=gradient(x0(1),x0(2)); %gradient of f at the initial point

disp(['initial point: (',num2str([xnew(1)]),',',num2str([xnew(2)]),')']);
disp(['iteration ',num2str(0),', the current point is:(',num2str([xnew(1)]),',',num2str([xnew(2)]),')']);


i=0;
while (norm(gradnew,2)>1e-9)
%% count the iterations
i=i+1;
  %% current point
  x=xnew; 
  
  %%find the optimal step parameter 
  tempf=@(lambda)cos(xnew(1)-lambda*gradnew(1))*cos(xnew(2)-lambda*gradnew(2));
  tstep=fminbnd(tempf, -4, 4);

  %%compute the new point using the current one and the multiple of the
  %%gradient vector
  xnew=x-tstep*gradnew';
  
  %%compute the gradient of f at the new point 
  gradnew=gradient(xnew(1),xnew(2));
  
  %%graph the new point and the line path between the previous and the new
  %%points
  hold on
  plot(xnew(1),xnew(2),'xk');
  x1=[x(1),xnew(1)];
  y1=[x(2),xnew(2)];
  plot(x1,y1)
  
  disp(['iteration ',num2str(i),', the current point is:(',num2str([xnew(1)]),',',num2str([xnew(2)]),')'...
            '  with step sizing: ', num2str(tstep) ]);
        if i>100 
            break;
        end
end
hold off