% Script 1 for Week 3


fun = @sin; % function
fun(pi)
x0 = 3; % initial point
xcoord1 = fzero(fun,x0)

[xcoord ycoord]= fzero(fun,[10,20])



% The fminbnd command in MATLAB can be used to find the 
% value of a function that is a (possibly local) minimum of the function. 
% The command can only find one minimum at a time and can only 
% find minima based on one variable at a time. If there is a 
% single local minimum over the domain, fminbnd should find it. 
% If there are several, it should find one of them.

t=linspace(2,7,1000);
y=sin(t.*t);
plot(t,y)


g = fminbnd(@(x) sin(x.*x),2,7)


% fminsearch tries to find the minimum of a 
% function of one or several variables.
% The seed value must be a vector or single number.
 
 w = fminsearch(@(x) x^2-7*x+1, 0)
 
banana = @(x)100*(x(2)-x(1).^2).^2+(1-x(1)).^2;
[xval,fval] = fminsearch(banana,[-1.2, 1])
 
[x,y] = meshgrid([-2:.2:2]);
Z = 100*(y-x.^2).^2+(1-x).^2;
surf(x,y,Z)
colorbar


% More fun with plots. Visualizing four dimensions with color. 
% If v is temperature as a function of position in space 
% this is a way of visualizing it. 
% The color paints the slices according to the value of v.

[x,y,z] = meshgrid(-2:.2:2,-2:.25:2,-2:.16:2);
v = x.*exp(-x.^2-y.^2-z.^2);
xslice = [-1.2,.8,2]; 
yslice = 2; 
zslice = [-2,0];
slice(x,y,z,v,xslice,yslice,zslice)
colormap hsv
colorbar


% Below are the scripts for a couple of test function M-files
% They illustrate how to find minima over irregular regions.

%*******************************************
function z=testfunc1(x,y)

if x.^2+y.^2>4 z=Inf;

elseif (y<0) & (x>0) & (y<-x) z=Inf;

elseif (y<0) & (x<=0) z=Inf;

else z=exp(-y).*cos(5.*x+y);

end
%*******************************************
%*******************************************
function z=testfunc2(x,y)

if 3*pi/4 < x z=Inf;

elseif pi/4 > x z=Inf;

elseif 0 > y z=Inf;

elseif x+y > pi z=Inf;

else z = 2./(2+sin(y))+3./(2+sin(x));

end
%*******************************************
clear
[x,y] = meshgrid([-3:.05:3]);

for k=1:length(x);
for w=1:length(y);
Z1(k,w) = testfunc1(x(k,w), y(k,w));
end
end

surf(x,y,Z1)
colorbar
shading('interp')
surfc(x,y,Z1)

% tricky issue:

% fminsearch requires that the functions it examines 
% have a single variable, but that variable can be a vector.

[xval,fval] = fminsearch(testfunc1,[-1.2, 1])

% fails because testfunc1 requires two inputs.

% But 
test1=@(x) testfunc1(x(1),x(2))
[a b]=fminsearch(test1,[-1.2 1])

% works, producing the x,y coordinate of the local min 
% (not a global min!!) and the actual minimum value there.

for k=1:length(x);
for w=1:length(y);
Z2(k,w) = testfunc2(x(k,w), y(k,w));
end
end


surfc(x,y,Z2)
colorbar
shading('interp')

% Find approximate location of min by examining graph: there is a 
% data cursor you can select near the top of a figure to get coordinates.


[xval,fval] = fminsearch(testfunc2,[1, 1])

% fails because testfunc2 requires two inputs.

% But 

test2=@(x) testfunc2(x(1),x(2))
[a b]=fminsearch(test2,[1 1])

% works, producing the x,y coordinate of the local min in a and the 
% actual minimum value there in b.