A new approach to numerical differentiation and integration

Abstract

Some new formulae for numerical differentiation and integration are derived by using interpolatory subdivision algorithms. These interpolatory subdivision algorithms are originally designed for the generation of smooth curves. The main advantage of these numerical formulae is that they produce better numerical results if the data comes from functions with fractal-like derivatives. The main disadvantage of these formulae is that they normally do not have the best approximation orders. By using different interpolatory subdivision algorithms, higher order approximation formulae can be obtained. Some numerical examples are given to compare these formulae with the traditional high accuracy formulae.