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.
