I am looking for practical error estimates for Newton-Cotes Quadrature rules.

Most books on numerical methods I have found mainly deal with theoretical error bounds/estimates for the respective methods.

Despite an extensive (but perhaps misdrirected) search I found almost nothing dealing with error estimation when automatic computation is concerned.

Example: The error of the Simpson-Rule is given by: $Ch^5 f^{(4)}(\xi)$ where $C$ is a constant and $\xi \in [a,b]$. Obviously $\xi$ usually is not known. One could certaily choose the maximum of $f^{(4)}$ to get an upper bound but this can be tricky to determine on the run.

This is why I am wondering: what are the best practices for estimating Integration-Error in numerical algorithms ? (above all I am interested in the ones concerning newton-cotes-quadrature)

• This was a standard topic for numerical computation manuals of the 1970s and 1980s. Maybe you aren't looking back far enough. One method that I recall was to use several values of $h$ and compare the estimates. Commented Jul 27, 2013 at 1:57
• still I wondering why hardly any traces of it remained in todays literature. Adaptive Quadrature is explained everywhere. Unfortunately the authors fail to mention how to actually implement it Commented Jul 27, 2013 at 6:04
• en.wikipedia.org/wiki/Romberg%27s_method Commented Jul 28, 2013 at 22:00
• @BrendanMcKay thanks to you comment I was at least able to determine that papers dealing with the subject exist. Fortunately I found a "fresh" one: arxiv.org/abs/1003.4629 Commented Jul 29, 2013 at 6:57