A recursive algorithm for optimizing differentiation

In this paper a recursive algorithm will be introduced to improve the differentiation method proposed by Hasan et al. (2012). Their algorithm is based on the undetermined coefficient method and uses the Taylor series expansion and Vandermonde matrix inverse to calculate weighting coefficients with a complexity of O(n4). Our method reduces time complexity from O(n4)  to  O(n3). Moreover, we obtain a substantial optimality when the required degree and the order of accuracy increase. Finally, the implementation of the proposed method follows, and experimentations prove the validity of the algorithm and compare previous and new methods in terms of execution time. Besides, the combination of our optimization and parallel processing provides valuable results in real world applications. Particularly, QRS detection in ECG signal processing has been considered, and we have shown that how features of our method could be used in order to increase accuracy and speed for this application.