A variant of Cauchy's method with accelerated fifth-order convergence

Abstract

We suggest an improvement to the iteration of Cauchy's method viewed as a generalization of possible improvements to Newton's method. Two equivalent derivations of Cauchy's method are presented involving similar techniques to ones that have been proved successfully for Newton's 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.