%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   SCRIPT              %%%%%%%
%%%%%%  phi.m                %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function y=phi(h)
y=(exp(1+h)-exp(1-h))/(2*h);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   SCRIPT              %%%%%%%
%%%%%%  der_first_order.m    %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all, 
m=40, format long e
n=1:m; h=1./2.^n;
e=exp(1);
err= (exp(1+h)-e)./h - e;
[h' err']
figure(1), plot(h,err)
figure(2), plot(log(h),log(abs(err)))
%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   SCRIPT              %%%%%%%
%%%%%%  der_sec_order.m      %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
format long
m=40
n=1:m; h=1./2.^n;
e=exp(1);

err_2 = phi(2/(2.^n))-e; 

[h'  err_2']
figure(1), plot(h,err_2)
figure(2), plot(log(h),log(abs(err_2)))
%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%   SCRIPT              %%%%%%%
%%%%%%  Richardson.m         %%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all, N=8, h=1
format long, e=exp(1)
for i=1:N,
    a(i)=phi(h/(2^i));
    error(i)=a(i)-e;
end
[a' error'], pause 
V=1
for j=1:N-1,
    V=4*V;
    for i=N:-1:j+1,
        a(i)=a(i) + (a(i)-a(i-1))/(V-1);
        error(i)=a(i)-e;
    end
end
error'
%%%%%%%%% END %%%%%%%%%%%%%%%%%%%%%%%%