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. – Brendan McKay Jul 27 '13 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 – Boldwing Jul 27 '13 at 6:04
• en.wikipedia.org/wiki/Romberg%27s_method – Federico Poloni Jul 28 '13 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 – Boldwing Jul 29 '13 at 6:57