Romberg's method

Remarks

Each column is a Richardson extrapolation of the previous one, which removes the term $h^{2m}$ from the error expansion. If $f$ is smooth enough, the error of $R(n, m)$ is $O(h_n^{2m + 2})$ as $n \to \infty$. The values $R(n, 0)$ can be computed recursively: $R(n, 0) = \frac12 R(n - 1, 0) + h_n \sum_{k=1}^{2^{n-1}} f(a + (2k - 1) h_n)$, so no function value is computed twice.

Example: $f(x) = x^2$ on $[0, 1]$. Then $R(0, 0) = \frac12 (0 + 1) = \frac12$ and $R(1, 0) = \frac12 \big( \frac12 \cdot 0 + \frac14 + \frac12 \cdot 1 \big) = \frac38$. So $R(1, 1) = \frac38 + \frac{3/8 - 1/2}{3} = \frac13$, which is the exact integral, as Simpson's rule is exact for quadratics.

Non-example: replacing $4^m - 1$ by $2^m - 1$ gives a different table that does not cancel the even powers of $h$, so it is not Romberg's method.