Abstract
A novel family of open Newton-Cotes (ONC) formulas is devised for evaluating the definite integrals. The new family is developed by using the Heronian mean in the first-order derivatives of the integrand within the interval [a, b]. The devised Heronian mean derivative-based quadrature rules (HRMDONC) achieve two orders of accuracy enhancement over the conventional ONC quadrature rules. These formulas are derived using the idea of degree of precision. Theorems regarding the degree of precision and order of accuracy are also derived along with the local and global error terms. In addition, the computational order of accuracy of each method is computed confirming the theoretical results. Computational cost and absolute error drops are also determined for three different integrals from the literature which demonstrate the superiority of the proposed HRMDONC methods over the classical ONC.
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