Richardson extrapolation

Remarks

The last claim is a direct computation: in $t^k A(h/t) - A(h)$ the terms $t^k c (h/t)^k$ and $c h^k$ cancel, and $t^k A^* - A^* = (t^k - 1)A^*$. Example: for a function $f$ with five continuous derivatives, the central difference $A(h) = \frac{f(x+h) - f(x-h)}{2h}$ satisfies $A(h) = f'(x) + \frac{f'''(x)}{6} h^2 + O(h^4)$. With $k = 2$ and $t = 2$, the extrapolation $\frac{4A(h/2) - A(h)}{3}$ approximates the derivative $f'(x)$ with error $O(h^4)$. Non-example: if the true error is $A(h) = A^* + h$ but one uses $k = 2$, then $R(h) = A^* + \frac{h}{3}$, which is still of first order; the exponent $k$ must be the true leading exponent. The procedure can be repeated with the next exponent of the error expansion, which gives a triangular table of extrapolated values.