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$$f''(x) \thickapprox\dfrac{1}{2h^2}[f(x+2h) - 2f(x) + f(x - 2h)]$$

I'm supposed to be deriving the above formula and establish an error formula in using them.

This is one of a series of problems like this, and I'm not quite too sure on how to get started on this. (This is in a chapter of Estimating Derivatives and Richardson Extrapolation)

Any tips/pointers?

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Is the denominator $(2h)^2$? –  Maesumi Jun 8 '13 at 3:27
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2 Answers

Taylor expansion does the trick:

$$\begin{align*} f(x+2h)&=f(x)+2hf^\prime(x)+2h^2 f^{\prime\prime}(x)+\frac43 h^3 f^{(3)}(x)+\cdots\\ f(x-2h)&=f(x)-2hf^\prime(x)+2h^2 f^{\prime\prime}(x)-\frac43 h^3 f^{(3)}(x)+\cdots \end{align*}$$

Combine these two appropriately to remove the term containing $f^\prime(x)$, and move everything to the left such that the first term of the right hand side is $f^{\prime\prime}(x)$. The term(s) after that should be your error estimate.

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Hint: $f''(x) \simeq \dfrac{f'(x+h)-f'(x-h)}{2h}.$ Try to replace $f'$ by $f$ in the definition.

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