Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Some Generalized Error Inequalities and Applications | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 845934 15 pages doi 2008 845934 Research Article Some Generalized Error Inequalities and Applications Fiza Zafar1 and Nazir Ahmad Mir2 1 Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University Multan 60800 Pakistan 2 Department of Mathematics COMSATS Institute of Information Technology Plot no. 30 Sector H-8 1 Islamabad 44000 Pakistan Correspondence should be addressed to Fiza Zafar fizazafar@ Received 4 February 2008 Revised 30 May 2008 Accepted 29 July 2008 Recommended by Sever Dragomir We present a family of four-point quadrature rule a generalization of Gauss-two point Simpson s 3 8 and Lobatto four-point quadrature rule for twice-differentiable mapping. Moreover it is shown that the corresponding optimal quadrature formula presents better estimate in the context of four-point quadrature formulae of closed type. A unified treatment of error inequalities for different classes of function is also given. Copyright 2008 F. Zafar and N. A. Mir. 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. 1. Introduction We define b I f f x dx. a The problem of approximating I f is usually referred to as numerical integration or quadrature 1 . Most numerical integration formulae are based on defining the approximation by using polynomial or piecewise polynomial interpolation. Formulae using such interpolation with evenly spaced nodes are referred to as Newton-Cotes formulae. The Gaussian quadrature formulae which are optimal and converge rapidly by selecting the node points carefully that need not be equally spaced are investigated in 2 . In 3-5 the quadrature problem in particular the investigation of error bounds of Newton-Cotes formulae namely the mid-point trapezoid .
