Local Convergence and Dynamics of a Family of Iterative Methods for Multiple Roots of Nonlinear Equations

The local convergence of a family of iterative methods based on a parameter θ for multiple roots of nonlinear equations is established. It is established under the assumption that the derivative g(m+ 1) of the function g satisfies the Holder continuity condition. The existence–uniqueness theorem along with a posteriori error bounds are provided. The R-order convergence is shown to be equal to (2 + p), where p ∈ (0, 1]. The well-known methods of Chebyshev (θ = 0) and of Osada (θ = 1) belong to the family. A number of numerical examples are worked out. The radii of convergence balls for different values of the parameter θ are compared and tabulated. To the best of our knowledge, our work is the first attempt to establish the local convergence of a family of iterative methods based on a parameter θ for computing multiple roots of nonlinear equations. Moreover, its dynamical study is investigated and the parameter plane is constructed to find the best value of the parameter θ. The values of θ are investigated according to the character of the strange fixed point and the multiplicity of the root.