I recently posted a question about computing error estimation of central difference formula:
$$f'(x)\approx D(x)=\frac{f(x+h)-f(x-h)}{2h}$$
It turns out that:
$$f'(x)=D(x)-\frac{h^2}{3}f'''(x)+\dots$$
However, in my case, I can't symbolically calculate $f'''(x)$. So my question is, is there any other, even less accurate, method to calculate $\epsilon$ which will always hold:
$$\epsilon\ge|D(x) - f'(x)|$$
Why I need it? I, just for fun, wrote a really simple program for symbolic differentiation and need to check that my output seems to be correct.
Lets say that $\hat{f'}(x)$ is symbolic differentation computed by my program. Sometimes $\hat{f'}(x)=f'(x)$, sometimes $\hat{f'}(x)\ne f'(x)$. I want to check that $\hat{f'(x)}$ is close to $D(x)$ to validate my program. Unfortunatelly, it's vary hard to algorithmically say what "close to" mean.
Maybe I'm missing something, but why can you not just calculate the quantity $D(x) - f'(x)$ explicitly for a series of test points? Of course, this will not guarantee that your bound will always be smaller than $\varepsilon$ but it should suffice for practical purposes. In general, the error bound that you derived doesn't look particularly helpful if $f'''$ is unknown or you don't know how to bound it explicitly. As it is an a-priori bound, it is more useful in the sense of knowing the rate at which the error goes to zero rather than for practical purposes.