The first kind Chebyshev–Newton–Cotes quadrature rules (semi-open type) and its numerical improvement

摘要

One of the less-known integration methods is the weighted Newton–Cotes quadrature rule of semi-open type, which is denoted by∫a=x0b=xn+1=x0+(n+1)hf(x)w(x)dx≃∑k=0nwkf(x0+kh),where w(x) is a weight function on [a, b] and h=b-an+1 is a positive value. There are various cases for w(x) that one can use. Because of the special importance of the weight function of Gauss–Chebyshev quadrature rules in the numerical analysis, i.e. w(x)=11-x2, we consider this function as the main weight. Hence, in this paper, we face with the following formula:∫-1+1f(x)1-x2dx≃∑k=0nwkf-1+2kn+1,which has the precision degree is n + 1 for even n’s and n for odd n’s. In this paper, we consider bounds of above integration formula as two additional variables to reach a nonlinear system that numerically improves the precision degree up to n + 2. In this way, sevral examples are given to show the numerical superiority of our approach.