A study on the average case error of composite Newton-Cotes quadratures. (Q1432806)

The authors study the integration problem using the composite Newton-Cotes quadrature to approximate a definite integral in the average case setting, present the average case error and show that it is minimal: Let \([0,1]\) be divided into \(n\) equal subintervals and each subinterval be divided into \(p-1\) equal subintervals. Then the average case errors of composite Newton-Cotes quadrature \(A\) with information \(N\) is \[ e^{avg}(A,N)=\begin{cases} \Theta\left(\tfrac{1}{n^p}\right), & \text{if }r\geq p,\\ \Theta\left(\tfrac{1}{n^{r+1}}\right), & \text{if }r<p\end{cases} \] for the set \(C^r[0,1]\).