NUMERICAL INTEGRATION BY NEWTON-COTES AND GAUSSIAN METHODS

Abstract

The main purpose of this study is to find approximate solution of definite integral which cannot be easily evaluated analytically. This is because, in order to use the fundamental theorem of calculus, anti-derivative of the function must be found, and that is not always possible or may be very difficult. The important preliminary concepts, examples, theorems, and some applications in different fields were presented to make the concept of both methods more clear. Many methods are available for approximating those integrals to the desired precision in numerical integration. This project is concerned with two quadrature methods (NCFs and GQ rule). Under NCFs, two type of NCF (CNCF and ONCF) were presented and their improved formulas CNCFMPD and ONCFMPD, which uses the derivative value at the midpoint, were proposed. The computational cost for these methods was analyzed for several examples and numerical results were compared and we showed that proposed scheme increase two order of precision than the existing. All NCFs were used values of the functions at equally-spaced points, but in GQ, nodes are zeros of certain orthogonal polynomials (Legendre, Leguerre, Hermite, and Chebyshev polynomials). Thus we relate GQ with those polynomials. Furthermore, three-term recurrence relation of those orthogonal polynomials, which used to obtain weights and nodes of GQ from Jacobi matrix are developed. This was gave a huge advantage in calculating integrals numerically by GQ rule. By comparisons of both methods we obtained that GQ is more accurate. Most computations were performed using MATLAB codes. Moreover, this study was suggested further study for searching improved GQ and NCFs.