Secant-Type Iterative Classes for Nonlinear Equations with Multiple Roots

General-purpose iterative methods for solving nonlinear equations provide approximations to solving problems without closed-form solutions. However, these methods lose some properties when the problems have multiple roots or are not differentiable, in which case specific methods are used. However, in most problems the multiplicity of the root is unknown, which reduces the range of methods available to us. In this work we propose two iterative classes with memory for solving multiple-root nonlinear equations without knowing the multiplicity. One of the proposals includes derivatives, but the other is derivative-free, obtained from the previous one using divided differences and a parameter in its iterative expression. The order of convergence of the proposed schemes is analyzed. The stability of the methods is studied using real dynamics, showing the good behavior of the methods. A numerical benchmark confirms the theoretical study.