Authors : J.O. Fatokun and H.K. Oduwole
Abstract: This study concerns the derivation of higher order Newton-cotes closed integration using Lagrange�s interpolation polynomial functions. In literature, the conventional way of deriving these integration formulae is by making the xi equally spaced, so that there are only n+1 parameters ai to choose. The coefficients are then fixed either by using finite difference results or by considering it as a general formula with various coefficients which must be fixed. William presents coefficients for orders p = 1, 2,...8 . In this research, an extension of this was carried out for orders p = 9, 10, 11. Ralston gave a detailed mathematical treatment of the error analysis for the known Newton Cotes formulae of orders p=1,2,�,8. These manipulations are extremely cumbersome since it involves n+1 conditions and hence solving a large system of algebraic equations. This led to the new approach of using the Lagrange�s interpolation functions to derive the Newton Cotes Closed formulae of orders p = 1,2,�,11. A generalized formula was developed for order p=k. These methods were applied to some integrals such as exponential function and trigonometric functions. The numerical results show that higher order methods are needed for equally spaced intervals to increase the accuracy of the general Newton-cotes closed formulae.
J.O. Fatokun and H.K. Oduwole , 2007. A New Approach for the Derivation of Higher-Order Newton-Cotes Closed Integration Formulae . Journal of Engineering and Applied Sciences, 2: 1460-1464.