%  This script file runs the "Rates of Convergence" experiment on Page 49.
%  In class we said that the size of the derivative at the fixed point
%  determines how quickly the orbit converges, and here we test that.

%
%  The functions are labeled as Fa, Fb, ..., Fi, Fj as on page 49.

Fa=inline('x.^2+0.25');
Fb=inline('x.^2');
Fc=inline('x.^2-0.24');
Fd=inline('x.^2-0.75');

Fe=inline('0.4.*x.*(1-x)');
Ff=inline('x.*(1-x)');
Fg=inline('1.6.*x.*(1-x)');
Fh=inline('2.*x.*(1-x)');
Fi=inline('2.4.*x.*(1-x)');
Fj=inline('3.*x.*(1-x)');

%
%  CHANGE THESE DEPENDING ON WHAT GROUP YOU'RE USING (the ones showing are
%  for the first grouping, found analytically).
fixedpts=[1/2,0,-0.2,-1/2];


%The procedure is to begin with x=0.2 and iterate until the function
%reaches a fixed point (actually, we will measure the distance between two
%iterates to see if it is very very small).

x=0.2;
MaxIterates=100000;  %Bail out if we do too many iterations
SmallNumber=10^(-5);

%******************************************************************
%  This section just for fun- Get cobweb diagrams for the first 4
%
xa=-1;xb=1;
figure(1)
subplot(2,2,1);
cobweb(Fa,x,10,xa,xb);
subplot(2,2,2);
cobweb(Fb,x,10,xa,xb);
subplot(2,2,3);
cobweb(Fc,x,10,xa,xb);
subplot(2,2,4);
cobweb(Fd,x,10,xa,xb);
%
%************End of graphics section********************************

for j=1:MaxIterates
    x=Fd(x);             %YOU WILL NEED TO CHANGE THE FUNCTION NAME HERE
    Err1=abs(fixedpts(4)-x);  %CHANGE FIXED POINT HERE if necessary
    
    if Err1<SmallNumber  
        fprintf('Very close to fixed point %f in %d iterations\n',x,j);
        break
    end
 
end
   if j==MaxIterates
       fprintf('Went over maximum number of iterations- No convergence yet\n');
       fprintf('Current distance to fixed point is %f\n',Err1);
   end