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On numerical improvement of open Newton–Cotes quadrature rules. (English) Zbl 1088.65022

Summary: We discuss the numerical improvement of the open Newton-Cotes integration rules that are in forms of

a=x -1 b=x n+1 =x -1 +(n+2)h f(x)dx k=0 n B k (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 numerical tests are given to show the numerical superiority of our idea with respect to the usual open Newton-Cotes integration rules.

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