An Adaptive Hybrid Method for Numerical Integration with Error Control

Abstract

One of the core tools of scientific computation is numerical integration. The classical quadrature techniques like the Trapezoidal and the Simpson rule are very popular but they might be inefficient when the smoothness of the functions to be integrated is variable. This paper describes an adaptive hybrid quadrature which dynamically changes the quadrature rule to use a set of quadrature rules according to local error estimation. The technique also has an error control measure to make the procedure accurate with minimum computing efforts. The convergence and stability of the proposed approach is confirmed by a theoretical analysis, and the efficiency and robustness of the proposed approach are proved using numerical experiments of standard test functions. The findings reveal that there is a tremendous increment in the accuracy of the results when compared to the classical fixed-step quadrature schemes.