Some remarks of the trapesoid rule in numerical integration
type: Item
instance of: scholarly article
title: Some remarks of the trapesoid rule in numerical integration
zbMATH Open document ID: 0967.41019
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2001-09-04T00:00:00Z
review text: 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 \(f''\).
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 1484557
zbMATH Keywords: numerical integration
zbMATH Keywords: trapezoidal rule
author: Silvestru Sever Dragomir
author: P. Cerone
author: Anthony Sofo
reviewed by: Doron S. Lubinsky
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1484557