The second kind Chebyshev–Newton–Cotes quadrature rule (open type) and its numerical improvement

摘要

The weighted Newton–Cotes quadrature rules of open type are denoted by∫a=x-1b=xn+1=x-1+(n+2)hf(x)w(x)dx≃∑k=0nwkf(x-1+(k+1)h),where w(x) is a positive function and h=b-an+2 is the step size. Various cases can be selected for the weight function of the above formula. In this paper, we consider w(x)=1-x2 as the main weight function and study the general formula:∫-1+1f(x)1-x2dx≃∑k=0nwkf-1+2(k+1)(n+2).The precision degree of the above formula is n + 1 for even n’s and is n for odd n’s but if one considers its upper and lower bounds as two additional variables, a nonlinear system will be derived whose solution improves the precision degree of above formula up to degree n + 2 numerically. In this way, some examples are given to show the numerical superiority of our idea.