A close look at Newton-Cotes integration rules

Abstract

Newton--Cotes integration rules are the simplest methods in numerical integration. The main advantage of using these rules in quadrature software is ease of programming. In practice, only the lower orders are implemented or tested, because of the negative coefficients of higher orders. Most textbooks state it is not necessary to go beyond Boole's 5-point rule. Explicit coefficients and error terms for higher orders are seldom given literature. Higher-order rules include negative coefficients therefore roundoff error increases while truncation error decreases as we increase the number of points. But is the optimal one really Simpson or Boole?

In this paper, we list coefficients up to 19-points for both open and closed rules, derive the error terms using an elementary and intuitive method, and test the rules on a battery of functions to find the optimum all-round one.

Keywords

• Quadrature,Newton--Cotes,Truncation Error,MATLAB

Kaynakça

1. E. Sermutlu, Comparison of Newton--Cotes and Gaussian methods of quadrature, Applied Mathematics and Computation, 171 (2005) 1048--1057
2. T.H.Fay and P.G.Webster, Lagrange interpolation and Runge's example, International Journal of Mathematical Education in Science and Technology, 27(6) (1996) 785--795
3. M. El-Mikkawy, A unified approach to Newton--Cotes quadrature formulae, Applied Mathematics and Computation, 138 (2003) 403--413
4. M. El-Mikkawy, On the error analysis associated with the Newton--Cotes formulae, International Journal of Computer Mathematics, 79(9) (2002) 1043--1047
5. M. Dehghan, M. Masjed-Jamei, M.R. Eslahchi, On numerical improvement of closed Newton--Cotes quadrature rules, Applied Mathematics and Computation, 165 (2005) 251--260
6. M. Dehghan, M. Masjed-Jamei, M.R. Eslahchi, On numerical improvement of open Newton--Cotes quadrature rules, Applied Mathematics and Computation, 175 (2006) 618--627
7. T.E.Simos, High--order closed Newton--Cotes trigonometrically - fitted formulae for long-time integration of orbital problems, Computer Physics Communications, 178 (2008) 199--207
8. M.~Abramowitz and I.A.~Stegun, Handbook of Mathematical Functions, Dover Publications, 1965

9. D.R. Hayes and L. Rubin, A proof of the Newton--Cotes quadrature formulas with error term, The American Mathematical Monthly, 77(10) (1970) 1065--1072
10. G.A. Evans, The estimation of errors in numerical quadrature, International Journal of Mathematical Education in Science and Technology, 25(10) (1994) 727--744
11. K.N. Melnik, R.V.N. Melnik, Optimal-by-order quadrature formulae for fast oscillatory functions with inaccurately given a priori information, Journal of Computational and Applied Mathematics, 110 (1999) 45--72
12. K.N. Melnik, R.V.N. Melnik, Optimal cubature formulae and recovery of fast-oscillating functions from an interpolational class, BIT, 41(4) (2001) 748--775
13. K.N. Melnik, R.V.N. Melnik, Optimal-by-accuracy and optimal-by-order cubature formulae in interpolational classes, Journal of Computational and Applied Mathematics 147 (2002) 233--262
14. E. Sermutlu, H.T. Eyyubo\~{g}lu, A new quadrature routine for improper and oscillatory integrals, Applied Mathematics and Computation 189 (2007) 452--461