Extended convergence analysis of optimal eighth order method for solving nonlinear equations

Abstract. To estimate the locally unique solutions of nonlinear systems in Banach spaces, the local convergence analysis of an optimal eighth order iterative technique is accomplished in this paper. The convergence analysis was carried out, in earlier studies, using the Taylor series expansions involving derivatives up to the ninth order. But, due to the fact that such higher order derivatives may not exist always, or laborious to evaluate for larger systems, the applicability of this technique remains limited. In this regard, we provide the convergence analysis using different approach which utilizes derivatives and divided differences only up to first order. The applicability of method is extended by computing upper bounds on the convergence region and prescribing set of conditions on the existence of unique solution in that region. Numerical examples are provided for the verification of theoretical results.