% this file determines the order and the constant C of the choosen discretization method

% n       Number of subdivisions of the intrvall[0,1]
% h=1/n   length of a stepp
% error   maximal difference of the exact solution u and its numerical approximation on the grid
% V       a vector used to compare graphically the behaviour of the error 

% Why do we express the error (in dependence of h) with loglog?

% error = C*h^order = C*exp(order*log(h))
% log(error) = log(C) + order*log(h)
% loglog(h,error) plots log(error) over log(h)
% => an even with
% incrementation   = order
% the value of the even at x=0  = log(C)

% the plot comparse with a of the same

disp('Enter the method as a string and put the second argument plot to 0')
disp('for example:')
f=input('`FpointDi1(n,0)`         :');
error=[];
for n=10:10:100
   error=[error,eval(f)];
end

% the different distances between the nodes
n=10:10:100;

% calculating the order of the choosen method
log_e=log(error);
log_n=log(n);
% order as average incrementation
Ord =(log_e(2:10)-log_e(1:9))./(-log_n(2:10)+log_n(1:9));
Ord = round(mean(Ord));

% V: vector to compare
V=n.^-Ord;

% calculating the constant C
C=mean(error./V);

% preparations for the plot
Ord = num2str(Ord);
C=num2str(C);
loglog(n,error,'r')
hold on
loglog(n,error,'r+')
loglog(n,V)
hold off

title('graphical determination of the order of the chosen method')
xlabel('n')
ylabel('error & V=vector to compare')
if str2num(C)>1
   text(0.7,0.8,'error','color','r','units','normalized')
   text(0.77,0.8,'= Cn','color','r','units','normalized')
   text(0.77,0.812,'~','color','r','units','normalized')
   text(0.82,0.83,'-order','color','r','units','normalized')
   text(0.1,0.2,'V =','color','b','units','normalized')      
   text(0.15,0.2,'n','color','b','units','normalized')      
   text(0.16,0.23,'-','color','b','units','normalized')      
   text(0.17,0.23,Ord,'color','b','units','normalized')      
else
   text(0.1,0.2,'error','color','r','units','normalized')
   text(0.17,0.2,'= Cn','color','r','units','normalized')
   text(0.17,0.212,'~','color','r','units','normalized')
   text(0.22,0.23,'-order','color','r','units','normalized')
   text(0.7,0.8,'V =','color','b','units','normalized')      
   text(0.75,0.8,'n','color','b','units','normalized')      
   text(0.76,0.83,'-','color','b','units','normalized')      
   text(0.77,0.83,Ord,'color','b','units','normalized')      
end
text(0.28,0.95,'method ','color','k','units','normalized')
text(0.42,0.95,'=','color','k','units','normalized')
text(0.45,0.95,f,'color','k','units','normalized')

text(0.3,0.9,'order ','color','k','units','normalized')
text(0.42,0.9,'=','color','k','units','normalized')
text(0.45,0.9,Ord,'color','k','units','normalized')

text(0.39,0.85,'C ','color','k','units','normalized')
text(0.42,0.85,'=','color','k','units','normalized')
text(0.45,0.85,C,'color','k','units','normalized')