Finding the Tolerance of Adaptive Quadrature Estimations

Question

So I'm learning about the Adaptive Quadrature "Algorithm" for estimating numerical integration and I am have trouble figuring out how you can approximate the error and actual value between two nodes. For clarification, the example at this link has an example problem where $|I_2 - I(f)|$ is approximately $(1/15) |I_2 - I_1|$. I am having trouble seeing where they got that from. Any help would be much appreciated. Thank!

Answer 1

The general method behind this is called Richardson extrapolation.

The error is of size $O(h^4)$, where $h$ is the length of the $n=1/h$ sub-intervals.. Extracting the dominating term, this means that the error has the form $$I(h)-I(0)=C·h^4+O(h^5).$$ Now we compare $I(h)$ and $$I(h/2)=I(0)+C·h^4/16+O(h^5)$$ Eliminating the leading error term this leads to $$ 16·I(h/2)-I(h)=16·I(0)-I(0)+O(h^5) $$ and thus $$ I(0)=\frac{16·I(h/2)-I(h)}{15}+O(h^5) $$ so that $$ I(h/2)-I(0)=\frac{I(h)-I(h/2)}{15}+O(h^5) $$

In your source, $I_1=I(h)$, $I_2=I(h/2)$ and $I(f)=I(0)$.