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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:

a b=a+nh f(x)dx k=0 n B k (n) 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.

MSC:
65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures