I've been using this to compute the first order derivative's value of a function $f$ in a given point: $$f'(x) = \frac{f(x+\epsilon) - f(x-\epsilon)}{2\epsilon}$$
For some $\epsilon = 0.0001$ or so. But when I try to use the same formula for higher order derivatives it gives odd results
$$f''(x) = \frac{f'(x+\epsilon) - f'(x-\epsilon)}{2\epsilon}$$ $$f''(x) = \frac{\frac{f(x+\epsilon+\epsilon) - f(x-\epsilon+\epsilon)}{2\epsilon} - \frac{f(x+\epsilon-\epsilon) - f(x-\epsilon-\epsilon)}{2\epsilon}}{2\epsilon}$$
What am I doing wrong here?
You use a $1,-2,1$ scheme
$$f''(x) = \frac{\frac{f(x+\epsilon+\epsilon) - f(x-\epsilon+\epsilon)}{2\epsilon} - \frac{f(x+\epsilon-\epsilon) - f(x-\epsilon-\epsilon)}{2\epsilon}}{2\epsilon} = \frac{f(x+h) - 2f(x)+f(x-h)}{h^2},$$
with $h=2\epsilon$, which is the first one here. Here is a list of alternatives.
Do you implement it with the additional two fractions and include "${}+\epsilon-\epsilon$"?
For what application and with program what do you use it?
Odd calculation of what kind or to what accuracy?