Cycling in the Newton‐Raphson Algorithm

Summary To better understand the Newton‐Raphson algorithm more attention should be given to cases of non‐convergence, in particular to cases that lead to cycling iterates. The values of cycling iterates have been shown to be crucial to separating regions of convergence and divergence. This paper summarizes what has been written about cycling iterates and provides further examples of the Newton‐Raphson algorithm cycling when being used to solve for the zeroes of f(z). The examples include families of functions for which an initial value can be selected to initiate a cycle of any given length preceded by any given number of non‐cycling values and families of functions for which the cycle length is fixed and does not depend on the initial value. In each example, the regions of convergence and divergence are also delineated.