Romberg Integration: accuracy

Question

I'm applying the Romberg method to numerically integrate a data set of equally space, numerically determined values. I would like some estimate of the uncertainty (or accuracy or error) in my answer. I have found two different approaches in the literature. One approach estimates the uncertainty by calculating the difference between the last elements in last two Romberg rows. The other looks at the difference between the last two elements in the last row. In my case, those results are different. Does anyone have suggestions/comments/references?

Answer 1

You have two results with error order $O(h^{2m})$, more precisely you know that the error is $C·h^{2m}+O(h^{2m+2})$. By comparing the results for step size $h$ and $2h$, you get \begin{align} y(h)&=y^*+C·h^{2m}+O(h^{2m+2})\\ y(2h)&=y^*+C·(2h)^{2m}+O(h^{2m+2})\\ \\ y^*&=\frac{2^{2m}·y(h)-y(2h)}{2^{2m}-1}+O(h^{2m+2})\\ \\ y(h)-y^*&=\frac{y(2h)-y(h)}{2^{2m}-1}+O(h^{2m+2}) \end{align} where the third formula is the computation of the last value in the Romberg table and the last the correct error estimate. I'd guess that the difference in your observed error estimates is about this factor of $4^m-1$, at least as long as $h^{2m+2}$ is still larger in magnitude than the machine precision.