Approximating function derivatives

[Albert Chan]

(01-24-2024 02:31 PM)Pekis Wrote:  It seems nice for f(x0) and f'(x0), acceptable for f"(x0), but looses precision for f"'(x0)

f(x0+h) = f(x0) + f'(x0)*h + f''(x0)*h²/2! + f'''(x0)*h³/2! + O(h⁴)

if we fit points f(x0±h), f(x0±2h), and estimate f(x0), error = O(h⁴)

Rewrite same taylor series, horner style

f(x0+h) = f(x0) + h*( f'(x0) + h/2*( f''(x0) + h/3*( f'''(x0) + O(h) )))

Estimated f'''(x0) has error = O(h), much worse.

(01-26-2024 08:33 AM)Thomas Klemm Wrote:  Even derivatives

\(
\begin{align}
f_{+1} + f_{-1} &= 2f(x_0) + {f}''(x_0) h^2 + \mathcal{O}(h^4) \\
f_{+2} + f_{-2} &= 2f(x_0) + 4{f}''(x_0) h^2 + \mathcal{O}(h^4) \\
\end{align}
\)

Let f_±k = f_k + f_-k,      where f_k = f(x0 + k*h)

Ignore O(h^4), solve for f''(x0), we have:

\(\displaystyle
{f}''(x_0) ≈ \left(-\frac{1}{3}f_{±1} + \frac{1}{3}f_{±2}\right) /\,h^2
\)

If we have the missing gap, (x0, f(x0)), f''(x0) can be estimated slightly better.

\(
\begin{align}
(f_{±1} - 2 f_{0}) &= {f}''(x_0)\; h^2 + \mathcal{O}(h^4) \\
(f_{±2} - 2 f_{0}) &= {f}''(x_0)\; (2h)^2 + \mathcal{O}((2h)^4) \\
\end{align}
\)

Instead of ignore error terms, this time we cancel them:

\( 16×(f_{±1} - 2 f_0) - (f_{±2} - 2 f_0) ≈ {f}''(x_0)\; h^2 × (16 - 4) \)

\(\displaystyle
{f}''(x_0) ≈ \left(-\frac{5}{2}f_0 + \frac{4}{3}f_{±1} - \frac{1}{12}f_{±2}\right) /\,h^2
\)