%  Demo for Derivative Aprroximation.
%
%  f(x) = 4 sqrt(x) - ln(x)   at   x=4
%  f'(4) = 0.75       exact value.

format long g
h = 10.^(-(1:16));  h = h';  % generate step sizes
x = 4+h;                     % calculate the locations 4+h
y = 4*sqrt(x) - log(x);      % compute f(4+h);
f4 = 8 - log(4);             % compute f(4)  once and for all
df = (y-f4)./h;              % compute (f(4+h)-f(4))/h  for our selection of h
relerr1 = (df-0.75)/0.75;     % compute the relative error for each h
display('forward differences')
[h df relerr1]                % display h, the approx derivative and the relative error
pause

%  same procedure, this time step back, i.e. use f(4-h)
x = 4-h;                     % calculate the locations 4+h
y = 4*sqrt(x) - log(x);      % compute f(4+h);
df = -(y-f4)./h;              % compute (f(4+h)-f(4))/h  for our selection of h
relerr2 = (df-0.75)/0.75;     % compute the relative error for each h
display('backward differences')
[h df relerr2]                % display h, the approx derivative and the relative error

pause

% graph the relative errors in a loglog plot

loglog(h,abs(relerr1),h,abs(relerr2),'--')
pause
display(' both error curves have approximately the same shape, optimum near h = 10^(-7)')
pause

%  improved formula

x = 4+h;                     % calculate the locations 4+h
y2 = 4*sqrt(x) - log(x);      % compute f(4+h);
x = 4-h;                     % calculate the locations 4-h
y1 = 4*sqrt(x) - log(x);      % compute f(4-h);
df = 0.5*(y2-y1)./h;              % compute (f(4+h)-f(4-h))/(2h)  for our selection of h
relerr3 = (df-0.75)/0.75;     % compute the relative error for each h
display('refined derivative approximations')
[h df relerr3]                % display h, the approx derivative and the relative error

loglog(h,abs(relerr3),h,abs(relerr1),'--');
display('new optimum near h = 10^(-5)')
pause