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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 $$\int_{a=x_{-1}}^{b=x_{n+1}= x_{-1}+(n+2)h} f(x)\,dx\simeq \sum_{k=0}^n B_k^{(n)} f(x_{-1}+ (k+1)h).$$ 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
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References:
[1] Atkinson, K.: An introduction to numerical analysis. (1989) · Zbl 0718.65001
[2] Babolian, E.; Masjed-Jamei, M.; Eslahchi, M. R.: On numerical improvement of Gauss-Legendre quadrature rules. Applied mathematics and computation 160, No. 3, 779-789 (2005) · Zbl 1062.65028
[3] Burden, R. L.; Faires, J. Douglas: Numerical analysis. (2001) · Zbl 0671.65001
[4] Conte, S. D.; De Boor, C.: Elementary numerical analysis--an algorithmic approach. (1981) · Zbl 0496.65001
[5] Davis, R.; Rabinowitz, P.: Methods of numerical integration. (1984) · Zbl 0537.65020
[6] Dehghan, M.; Masjed-Jamei, M.; Eslahchi, M. R.: On numerical improvement of closed Newton-cotes quadrature rules. Applied mathematics and computation 165, No. 2, 251-260 (2005) · Zbl 1070.65018
[7] Eslahchi, M. R.; Masjed-Jamei, M.; Babolian, E.: On numerical improvement of Gauss-lobatto quadrature rules. Applied mathematics and computation 164, No. 3, 274-280 (2005) · Zbl 1070.65019
[8] Eslahchi, M. R.; Dehghan, M.; Masjed-Jamei, M.: On numerical improvement of the first kind of Gauss-Chebyshev quadrature rules. Applied mathematics and computation 165, No. 1, 5-21 (2005) · Zbl 1079.65023
[9] Gautschi, W.: Numerical analysis: an introduction. (1997) · Zbl 0877.65001
[10] Isaacson, E.; Keller, H. B.: Analysis of numerical methods. (1966) · Zbl 0168.13101
[11] Krylov, V. I.: Approximate calculation of integrals. (1962) · Zbl 0111.31801
[12] M. Masjed-Jamei, M.R. Eslahchi, M. Dehghan, On numerical improvement of Gauss-Radua quadrature rules, Applied Mathematics and Computation, in press. · Zbl 1085.41023