A Perturbed Milne’s Quadrature Rule for n-Times Differentiable Functions with L^p-Error Estimates

Abstract

In this work, in spite of Milne’s recommendation using the three-point Newton–Cotes open formula (Milne’s rule) as a predictor rule and three-point Newton–Cotes closed formula (Simpson’s rule) as a corrector rule for 4-th differentiable functions with bounded derivatives. There is still a great need to introduce such formulas in other Lp spaces. Often, we need to approximate real integrals under the assumptions of the function involved. Because of that, the aim of this work is to introduce several Lp error estimates for the proposed perturbed Milne’s quadrature rule. Numerical experiments showing that our proposed quadrature rule is better than the classical Milne rule for certain types of functions are provided as well.