CMMSE: Jacobian-free vectorial iterative scheme to find several solutions simultaneously

Abstract

This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.