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On numerical improvement of closed Newton-Cotes quadrature rules. (English) Zbl 1070.65018
Summary: This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of: $$\int^{b= a+ nh}_a f(x)\,dx\simeq \sum^n_{k=0} B^{(n)}_k f(a+ kh).$$ 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.

65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures
Full Text: DOI
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