The semi-open Newton–Cotes quadrature rule and its numerical improvement

摘要

In this paper we discuss about numerical improvement of the semi-open Newton–Cotes integration rules that are in forms of:∫a=x-1b=xn+1=x-1+(n+1)hf(x)dx≃∑k=0nBk(n)f(x-1+(k+1)h).It is known that the precision degree of above formula is n + 1 for even n′s and is n for odd n′s. However, if the integral bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n + 2. In this way, some computational tests are given to show the numerical superiority of our approach with respect to the usual semi-open Newton–Cotes integration rules.