Iterative Adaptive Simpson and Lobatto Quadrature

In almost every standard book on numerics quadrature algorithms like the adaptive Simpson or the adaptive Lobatto algorithm are presented in a recursive way. The benefit of the recursive programming is the compact and clear representation. However, recursive quadrature algorithms might be transformed into iterative quadrature algorithms without major modifications in the structure of the algorithm.

We present iterative adaptive quadrature algorithm (adaptiveSimpson and adaptiveLobatto), which preserves the compactness and the clarity of the recursive algorithms (e.g. quad, quadv, and quadl). Our iterative algorithm provides a parallel calculation of the integration function, which leads to tremendous gain in run-time, in general. Our results suggest a general iterative and not a recursive implementation of adaptive quadrature formulas, once the programming language permits parallel access to the integration function.