Approximating function derivatives

[Pekis]

Here is a subject slightly extended from the book "La calculatrice scientifique" Vangeluwe/Glorieux, 1979:

The point is to approximate function derivatives (targets are f(x0) or f'(x0) or f"(x0) or f"'(x0)) with the help of neighboring points (at distance h and h/2 from both sides of x0)

So, knowing h, find a,b,c,d such as
a * f(x0 + h) + b * f(x0 - h) + c * f(x0 + h/2) + d * f(x0 - h/2) ~= (f(x0) or f'(x0) or f"(x0) f"'(x0)), ie each target needs its (a,b,c,d) set

In any case, with Taylor:
f(x0 + h) = f(x0) + h * f'(x0) + h^2/2! * f"(x0) + h^3/3! * f"'(x0) + ...
f(x0 - h) = f(x0) - h * f'(x0) + h^2/2! * f"(x0) - h^3/3! * f"'(x0) + ...
f(x0 + h/2) = f(x0) + h/2 * f'(x0) + h^2/(2^2*2!) * f"(x0) + h^3/(2^3*3!) * f"'(x0) + ...
f(x0 - h/2) = f(x0) - h/2 * f'(x0) + h^2/(2^2*2!) * f"(x0) - h^3/(2^3*3!) * f"'(x0) + ...

=> Solve system:
a + b + c + d = 1 if target f(x0), else 0
a - b + c/2 - d/2 = 1/h if target f'(x0), else 0
a + b + c/4 + d/4 = 2!/(h^2) if target f"(x0), else 0
a - b + c/8 - d/8 = 3!/(h^3) if target f"'(x0), else 0

Results:
If target f(x0): a = -1/6, b = -1/6, c = 2/3, d = 2/3
If target f'(x0): a = -1/(6*h), b = 1/(6*h), c = 4/(3*h), d = -4/(3*h)
If target f"(x0): a = 4/(3*h^2), b = 4/(3*h^2), c = -4/(3*h^2), d = -4/(3*h^2)
If target f"'(x0): a = 4/(h^3), b = -4/(h^3), c = -8/(h^3), d = 8/(h^3)

Here a the results for f(x)=ln(x), h=0.001, at x0=3, from Excel:


   

It seems nice for f(x0) and f'(x0), acceptable for f"(x0), but looses precision for f"'(x0)

Did you already know that method ?