Derivative-based closed Newton–Cotes numerical quadrature

Abstract

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 formula 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.