A variant of Cauchyʹs method with accelerated fifth-order convergence Original Research Article

We suggest an improvement to the iteration of Cauchyʹs method viewed as a generalization of possible improvements to Newtonʹs method. Two equivalent derivations of Cauchyʹs method are presented involving similar techniques to ones that have been proved successfully for Newtonʹs method. First, an adaptation of an auxiliary function that gives the new iteration function, and secondly, a symbolic computation that allows us to find the best coefficients with regard to the local order of convergence. The theoretical and computational order of convergence, for all functions tested, was five or more.