Generalized convergence conditions for the local and semilocal analyses of higher order Newton-type iterations

Abstract

In this work, some higher order Newton-type iterative methods are analyzed comprehensively in Banach spaces to estimate the locally unique solutions of systems of nonlinear equations. Being the Newton-type methods, these require only the first-order derivative computation. However, the convergence analysis is usually executed by utilizing the Taylor expansions which inherently include the assumptions on the existence of derivatives of higher order. Such assumptions certainly restrict their applicability. In this regard, the local and semilocal convergence analyses are developed by imposing the hypotheses only on the first order derivatives. In local analysis, the prime focus is to provide bounds on the domain of convergence along with estimating the error approximations of successive iterates. In semilocal analysis, the sufficient conditions are based on arbitrarily chosen initial approximation in a given domain that ensures the convergence of iterative sequence to a particular solution in that domain. The uniqueness of solution is further claimed by providing the sufficient criteria in the given domain. Lastly, the theoretical deductions are certified by testing results on some applied problems.