Burg, Clarence O. E. Derivative-based closed Newton-Cotes numerical quadrature. (English) Zbl 1246.65043 Appl. Math. Comput. 218, No. 13, 7052-7065 (2012). Summary: A new family of numerical integration formula of closed Newton-Cotes-type is presented, that uses both the function value and the derivative value on uniformly spaced intervals. Since there are more unknowns when using including derivative values in addition to function values, the order of accuracy of these numerical integration formula are higher than the standard closed Newton-Cotes formula. These new formulae are derived via the method of undetermined coefficients, based on the concept of the precision of the quadrature formula. The error terms are found in three different ways, using the concept of precision, using Taylor series expansions about the interval midpoint and using polynomial approximating functions, which is how the error terms for the standard closed Newton-Cotes formula were obtained.The concept of precision and the Taylor series methods yield the same error terms as the polynomial-based method, but there are certain unverifiable assumptions in their use. Quadrature formula using first derivatives at all points throughout the interval increase the order of accuracy to \(2n + 2\). Quadrature formula using the first derivatives at the endpoints of the interval obtain an increase of two orders of accuracy over the closed Newton-Cotes formula; while quadrature formula involving higher order derivatives result in substantially higher orders of accuracy, being \((D + 1)(n + 1)\) where \(D \) is the number of derivatives involved in the formula.The computational cost for these methods are analyzed for two different examples, showing the number of function and derivative evaluations necessary to reduce the error below a certain level. These two numerical examples are used to demonstrate that the theoretical order of accuracy is achieved by the numerical implementation of these formula. Additionally, a generalized Rolle’s Theorem with derivatives and a theorem for the error in the general interpolating polynomial including derivatives are provided in the appendix, which leads to the proof of some of the error terms in the derivative-based closed Newton-Cotes-type quadrature formula. Cited in 6 Documents MSC: 65D32 Numerical quadrature and cubature formulas 41A55 Approximate quadratures Keywords:numerical quadrature; closed Newton-Cotes integration; generalized Rolle’s theorem; polynomial interpolation; derivative-based quadrature; method of undetermined coefficients; Taylor series expansions; numerical examples PDFBibTeX XMLCite \textit{C. O. E. Burg}, Appl. Math. Comput. 218, No. 13, 7052--7065 (2012; Zbl 1246.65043) Full Text: DOI References: [1] Babolian, E.; Masjed-Jamei, M.; Eslahchi, M. R., On numerical improvement of Gauss-Legendre quadrature rules, Appl. Math. Comput., 160, 779-789 (2005) · Zbl 1062.65028 [2] Eslahchi, M. R.; Masjed-Jamei, M.; Babolian, E., On numerical improvement of Gauss-Lobatto quadrature rules, Appl. Math. Comput., 164, 707-717 (2005) · Zbl 1070.65019 [3] Esclahchi, M. R.; Dehghan, M.; Masjed-Jamei, M., On numerical improvement of the first kind Gauss-Chebyshev quadrature rules, Appl. Math. Comput., 165, 5-21 (2005) · Zbl 1079.65023 [4] Dehghan, M.; Masjed-Jamei, M.; Eslahchi, M. R., On numerical improvement of closed Newton-Cotes quadrature rules, Appl. Math. Comput., 165, 251-260 (2005) · Zbl 1070.65018 [5] Masjed-Jamei, M.; Eslahchi, M. R.; Dehghan, M., On numerical improvement of Gauss-Radau quadrature rules, Appl. Math. Comput., 168, 51-64 (2005) · Zbl 1085.41023 [6] Esclahchi, M. R.; Dehghan, M.; Masjed-Jamei, M., The first kind Chebyshev-Newton-Cotes quadrature rules (closed type) and its numerical improvement, Appl. Math. Comput., 168, 479-495 (2005) · Zbl 1084.65025 [7] Dehghan, M.; Masjed-Jamei, M.; Eslahchi, M. R., The semi-open Newton-Cotes quadrature rule and its numerical improvement, Appl. Math. Comput., 171, 1129-1140 (2005) · Zbl 1090.65033 [8] Dehghan, M.; Masjed-Jamei, M.; Eslahchi, M. R., On numerical improvement of open Newton-Cotes quadrature rules, Appl. Math. Comput., 175, 618-627 (2006) · Zbl 1088.65022 [9] Isaacson, E.; Keller, H. B., Analysis of Numerical Methods (1966), John Wiley and Sons: John Wiley and Sons New York · Zbl 0168.13101 [10] R. L. Burden, J. D. Faires, “Numerical Analysis, 8th ed.”, Thomson, Brooks/Cole, 2005.; R. L. Burden, J. D. Faires, “Numerical Analysis, 8th ed.”, Thomson, Brooks/Cole, 2005. · Zbl 0788.65001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.