= Solution
Write the computed function values as $f(x\pm h)+e_\pm$, where $|e_\pm|\le\delta$. The computed <central finite difference> $\widetilde d$ satisfies
$$
\boxed{|\widetilde d-d|=\left|\frac{e_+-e_-}{2h}\right|\le\frac\delta h,\qquad |\widetilde d-f'(x)|\le\frac{h^2}{6}+\frac\delta h.}
$$
The two contributions are respectively evaluation <absolute error> and <truncation error>. For $\delta>0$, <step-size choice for a noisy central derivative> minimizes their sum: its <derivative> is $h/3-\delta/h^2$, negative before and positive after
$$
\boxed{h=(3\delta)^{1/3},\qquad \min_{h>0}\left(\frac{h^2}{6}+\frac\delta h\right)=\frac{(3\delta)^{2/3}}2.}
$$
If $\delta=0$, the bound tends to zero as $h\downarrow0$ and has no positive minimizing step.
Back to article page
