Efficient numerical methods of Aitken type and their dynamics

In this paper, we derive some general Aitken type methods. These methods involve arbitrary methods of orders $p$ and $q$ which enable us to construct the method of any desired order. Further, it is shown that these methods can be combined with generalized secant method and as result, in the limiting case, the efficiency can be increased to $2$. We also discuss the stability of the iterative method with the help of basins of attraction in the complex plane. Some numerical examples are provided in support of the theoretical results.