Numerical estimation of second order derivative

Question

Given that y" = [y(x+h)-2y(x)+y(x-h)]/h^2 (central difference) with error = O(h^2). Find an expression for y" that has a smaller error than O(h^2).

Note: h is small (say 0 < h < 1). Thanks in advance

You will have to use more points in your approximation. For example you could use any $5$ distinct points of your choosing, and the error will be $O(h^{5 - 2}) = O(h^3)$ or better. — Mason, Dec 08 '20 at 04:49

score 0 · Answer 1 · answered Dec 08 '20 at 16:33

0

Thanks Mason. So I need to do Richardson's Extrapolation which gives an error of O(h^4). Check link below:

answered Dec 08 '20 at 16:33

William Kubin

You don't have to. WLOG, let $x = 0$. You can just pick any $n \geq 5$ distinct points $c_i h$ $(1 \leq i \leq n)$ of your choosing, and then solve an $n$ by $n$ linear system to get the coefficients $w_i$ $(1 \leq i \leq n)$ for the approximation $$f''(0) \approx \frac{1}{h^2}\sum_{i = 1}^{n}w_i f(c_ih).$$ The system is obtained from the fact that the approximation must be exact for $f = 1, x, x^2, \dots, x^{n - 1}$. – Mason Dec 08 '20 at 20:18

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