Some remarks of the trapesoid rule in numerical integration (Q1572793)

In this interesting paper, the authors use ideas from the theory of inequalities, to derive quadrature rules similar to the trapezoidal rule, but with smaller worst case error for twice differentiable functions. Let \(a=x_0< x_1 <x_2< \cdots<x_{n-1} <x_n=b\) be a partition of the interval \([a,b]\), and let \(h_j:= x_{j+1}- x_j\), \(0\leq j\leq n-1\). The associated trapezoidal rule may be written in the form \(A[f]:= {1\over 2}\sum^{n-1}_{j=0} [f(x_j)+ f(x_{j+1})] h_j\), and the classical error estimate is \[ \left|\int^b_af-A[f] \right |\leq{1\over 12} \|f''\|_{L_\infty [a,b]} \sum^{n-1}_{j=0} h^3_j. \] The authors show for example that if \(m:=\inf_{[a,b]} f''\) and \(M:= \sup_{[a,b]} f''\) then \[ \left|\int^b_af-A[f] -{M+m\over 24} \sum^{n-1}_{j=0} h^3_j \right|\leq{M-m\over 24} \sum^{n-1}_{j=0} h^3_j. \] They also derive error estimates for related rules and some possibly new error estimates for the ordinary trapezoidal rule involving \(L_p\) norms of \(f''\).