Richardson's Extrapolation

Question

Use Richardson's extrapolation to find a 3 point 2nd order approximation of f '(x).

I'm not sure how to go about to start this, i'm not the best when using richardson's extrapolation.

Answer 1

From the two-point forward difference formula, we have:

$$\tag 1 f'(x) \approx \dfrac{f(x+h)-f(x)}{h}$$

Using Richardson's, we replace $h$ by $\dfrac{h}{2}$ in $(1)$, yielding:

$$\tag 2 f'(x) \approx 2\dfrac{f(x+h/2)-f(x)}{h}$$

Subtracting $2 \times (2)$ from $(1)$ (this is to get the Richardson form) yields:

$$f'(x) \approx \frac{- f(x+h) + 4 f(x+h/2)- 3f(x) }{h} $$

This is a $3-$point, $2^{nd}$ order approximation.