Derivative-Based Midpoint Quadrature Rule

ABSTRACT

A new family of numerical integration formula is presented, which uses the function evaluation at the midpoint of the interval and odd derivatives at the endpoints. Because the weights for the odd derivatives sum to zero, the derivative calculations cancel out for the interior points in the composite form, so that these derivatives must only be calculated at the endpoints of the overall interval of integration. When using *N* subintervals, the basic rule which uses the midpoint function evaluation and the first derivative at the endpoints achieves fourth order accuracy for the cost of *N*/2 function evaluations and 2 derivative evaluations, whereas the three point open Newton-Cotes method uses 3*N*/4 function evaluations to achieve the same order of accuracy. These derivative-based midpoint quadrature methods are shown to be more computationally efficient than both the open and closed Newton-Cotes quadrature rules of the same order. This family of derivative-based midpoint quadrature rules are derived using the concept of precision, along with the error term. A theorem concerning the order of accuracy of quadrature rule using the concept of precision is provided to justify its use to determine the leading order error term.

Cite this paper

C. Burg and E. Degny, "Derivative-Based Midpoint Quadrature Rule,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 228-234. doi: 10.4236/am.2013.41A035.

C. Burg and E. Degny, "Derivative-Based Midpoint Quadrature Rule,"

References

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[5] M. Masjed-Jamei, M. R. Eslahchi and M. Dehghan, “On Numerical Improvement of Gauss-Radau Quadrature Rules,” Applied Mathematics and Computation, Vol. 168, No. 1, 2005, pp. 51-64. doi:10.1016/j.amc.2004.08.046

[6] M. R. Esclahchi, M. Dehghan and M. Masjed-Jamei, “The First Kind Chebyshev-Newton-Cotes Quadrature Rules (Closed Type) and Its Numerical Improvement,” Applied Mathematics and Computation, Vol. 168, No. 1, 2005, pp. 479-495. doi:10.1016/j.amc.2004.09.048

[7] M. Dehghan, M. Masjed-Jamei and M. R. Eslahchi, “The Semi-Open Newton-Cotes Quadrature Rule and Its Numerical Improvement,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 1129-1140. doi:10.1016/j.amc.2005.01.137

[8] M. Dehghan, M. Masjed-Jamei and M. R. Eslahchi, “On Numerical Improvement of Open Newton-Cotes Quadrature Rules,” Applied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 618-627. doi:10.1016/j.amc.2005.07.030

[9] C. O. E. Burg, “Derivative-Based Closed Newton-Cotes Numerical Quadrature,” Applied Mathematics and Computation, Vol. 218, No. 13, 2012, pp. 7052-7065 doi:10.1016/j.amc.2011.12.060

[10] The REXX Language Association. http://www.rexxla.org.

[11] http://www.rexxinfo.org.

[1] E. Babolian, M. Masjed-Jamei and M. R. Eslahchi, “On Numerical Improvement of Gauss-Legendre Quadrature Rules,” Applied Mathematics and Computation, Vol. 160, No. 3, 2005, pp. 779-789. doi:10.1016/j.amc.2003.11.031

[2] M. R. Eslahchi, M. Masjed-Jamei and E. Babolian, “On Numerical Improvement of Gauss-Lobatto Quadrature Rules,” Applied Mathematics and Computation, Vol. 164, No. 3, 2005, pp. 707-717. doi:10.1016/j.amc.2004.04.113

[3] M. R. Esclahchi, M. Dehghan and M. Masjed-Jamei, “On Numerical Improvement of the First Kind Gauss-Chebyshev Quadrature Rules,” Applied Mathematics and Computation, Vol. 165, No. 1, 2005, pp. 5-21. doi:10.1016/j.amc.2004.06.102

[4] M. Dehghan, M. Masjed-Jamei and M. R. Eslahchi, “On Numerical Improvement of Closed Newton-Cotes Quadrature Rules,” Applied Mathematics and Computation, Vol. 165, No. 2, 2005, pp. 251-260. doi:10.1016/j.amc.2004.07.009

[5] M. Masjed-Jamei, M. R. Eslahchi and M. Dehghan, “On Numerical Improvement of Gauss-Radau Quadrature Rules,” Applied Mathematics and Computation, Vol. 168, No. 1, 2005, pp. 51-64. doi:10.1016/j.amc.2004.08.046

[6] M. R. Esclahchi, M. Dehghan and M. Masjed-Jamei, “The First Kind Chebyshev-Newton-Cotes Quadrature Rules (Closed Type) and Its Numerical Improvement,” Applied Mathematics and Computation, Vol. 168, No. 1, 2005, pp. 479-495. doi:10.1016/j.amc.2004.09.048

[7] M. Dehghan, M. Masjed-Jamei and M. R. Eslahchi, “The Semi-Open Newton-Cotes Quadrature Rule and Its Numerical Improvement,” Applied Mathematics and Computation, Vol. 171, No. 2, 2005, pp. 1129-1140. doi:10.1016/j.amc.2005.01.137

[8] M. Dehghan, M. Masjed-Jamei and M. R. Eslahchi, “On Numerical Improvement of Open Newton-Cotes Quadrature Rules,” Applied Mathematics and Computation, Vol. 175, No. 1, 2006, pp. 618-627. doi:10.1016/j.amc.2005.07.030

[9] C. O. E. Burg, “Derivative-Based Closed Newton-Cotes Numerical Quadrature,” Applied Mathematics and Computation, Vol. 218, No. 13, 2012, pp. 7052-7065 doi:10.1016/j.amc.2011.12.060

[10] The REXX Language Association. http://www.rexxla.org.

[11] http://www.rexxinfo.org.