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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 492507, 10 pages
http://dx.doi.org/10.1155/2013/492507
Research Article

Midpoint Derivative-Based Closed Newton-Cotes Quadrature

1Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian 116024, China
2College of Air Traffic Management, Civil Aviation University of China, Tianjin 300300, China

Received 16 April 2013; Accepted 9 June 2013

Academic Editor: Luca Guerrini

Copyright © 2013 Weijing Zhao and Hongxing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A novel family of numerical integration of closed Newton-Cotes quadrature rules is presented which uses the derivative value at the midpoint. It is proved that these kinds of quadrature rules obtain an increase of two orders of precision over the classical closed Newton-Cotes formula, and the error terms are given. The computational cost for these methods is analyzed from the numerical point of view, and it has shown that the proposed formulas are superior computationally to the same order closed Newton-Cotes formula when they reduce the error below the same level. Finally, some numerical examples show the numerical superiority of the proposed approach with respect to closed Newton-Cotes formulas.