High Convergence Order Q-Step Methods for Solving Equations and Systems of Equations

Abstract

The local convergence analysis of iterative methods is important since it demonstrates the degree of difficulty for choosing initial points. In the present study, we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher order derivatives. This way we extend the applicability of these methods. The analysis includes the computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study.