function convergenceTaylor_vs_Euler()
% 
% y' = -y + 2 cos(t)
%   y(0) = 1
% exact y(t) = sin t + cos t


        
    clear, home, close all
    format compact
    format long

  

      finalTime = 10;
      Nv = [40 80 160 320 640 1280 2560 5120];
      
      
  for j=1:length(Nv)

       N = Nv(j);
      dt = finalTime/(N-1);

          tp = linspace(0,finalTime,N)';
          yn = zeros(N,1);  
       yn(1) = exactSolution(0); 

          yt = zeros(N,1);  
       yt(1) = exactSolution(0);
       

         
       t = 0; 
       for n=2:N
       
           yn(n) = yn(n-1) + dt*F(yn(n-1),t);
           
           yt(n) = yt(n-1) + dt*( -yt(n-1) + 2*cos(t) ) + 0.5*(dt^2)*( yt(n-1) - 2*cos(t) - 2*sin(t) );
           
           t = t + dt;  
           
        end  % for
        
        exact = exactSolution(tp);
        er(j) = abs( exact(N) - yn(N) );
        ert(j) = abs( exact(N) - yt(N) );
        
        DT(j) = dt;
 
       
   end % j
       
       loglog(DT,er,'b',DT,ert,'r')
       legend('Euler','Taylor','Location','SouthEast')
       
        for i=2:length(Nv);
          rho(i-1) = log(er(i-1)/er(i))/log(2);
          rhot(i-1) = log(ert(i-1)/ert(i))/log(2);
        end
        
        disp('Euler convergence rate: ')
        rho'
        disp('Euler errors: ')
        er'
        disp('Taylor convergence rate: ')
        rhot'
        disp('Taylor errors: ')
        ert'

    

% ----------- nested functions ---------------------------------

   function f = F(v,t)
      f = -v + 2.*cos(t);
   end
 
   function ex = exactSolution(t)
      ex = sin(t) + cos(t);
   end

% ----------------------------------------------------------

end