Local Convergence for an Efficient Eighth Order Iterative Method with a Parameter for Solving Equations Under Weak Conditions

We present a local convergence analysis of an efficient eighth order iterative method with a parameter for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Bi et al. (J Comput Appl Math 225:105–112, 2009) have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study.