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I've been using this to compute the first order derivative's value of a function $f$ in a given point: $$f'(x) = \frac{f(x+\epsilon) - f(x-\epsilon)}{2\epsilon}$$

For some $\epsilon = 0.0001$ or so. But when I try to use the same formula for higher order derivatives it gives odd results

$$f''(x) = \frac{f'(x+\epsilon) - f'(x-\epsilon)}{2\epsilon}$$ $$f''(x) = \frac{\frac{f(x+\epsilon+\epsilon) - f(x-\epsilon+\epsilon)}{2\epsilon} - \frac{f(x+\epsilon-\epsilon) - f(x-\epsilon-\epsilon)}{2\epsilon}}{2\epsilon}$$

What am I doing wrong here?

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Don't use previous derivatives to find higher-order ones. Use $f$ directly. See en.wikipedia.org/wiki/… –  lhf Mar 2 '12 at 14:47
    
Try this –  Inquest Mar 2 '12 at 14:54

1 Answer 1

up vote 1 down vote accepted

You use a $1,-2,1$ scheme

$$f''(x) = \frac{\frac{f(x+\epsilon+\epsilon) - f(x-\epsilon+\epsilon)}{2\epsilon} - \frac{f(x+\epsilon-\epsilon) - f(x-\epsilon-\epsilon)}{2\epsilon}}{2\epsilon} = \frac{f(x+h) - 2f(x)+f(x-h)}{h^2},$$

with $h=2\epsilon$, which is the first one here. Here is a list of alternatives.

Do you implement it with the additional two fractions and include "${}+\epsilon-\epsilon$"?

For what application and with program what do you use it?

Odd calculation of what kind or to what accuracy?

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Thanks! The odd results were because of me, but I fixed it and using the formula you suggested I managed to increase accuracy. I need it for a school project on "Newton polynomial" –  Diana Dcn Mar 2 '12 at 15:32

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