On numerical improvement of closed Newton-Cotes quadrature rules.

*(English)*Zbl 1070.65018Summary: This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of:

$${\int}_{a}^{b=a+nh}f\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\simeq \sum _{k=0}^{n}{B}_{k}^{\left(n\right)}f(a+kh)\xb7$$

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 its 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 above integration formula up to degree $n+2$. In this way, some numerical examples are given to show the numerical superiority of our approach with respect to usual Newton-Cotes integration formulas.