Extended convergence analysis of a generalized Newton-type iterative method: a unified local-semilocal approach

An extensive analysis is carried out on a higher-order Newton-type iterative method for estimating the locally unique solutions of nonlinear systems in Banach spaces. The iterative algorithm of given method does not require the computation of derivatives of order higher than one. Nonetheless, the convergence analysis is typically performed in previous studies with the assumption that higher order derivatives exist as well. These presumptions undoubtedly limit its application. In this context, this study discusses convergence of given method through the local and semilocal approaches wherein the conditions are imposed solely on first order derivative. Estimating the convergence domain along with the error estimates of iterations are the key ideas of local analysis. The semilocal analysis guarantees the convergence of iterates to a specific solution in the domain by establishing the sufficient conditions on the selection of initial estimate in that domain. It is further asserted that the solution is unique by providing the suitable criteria in the specified domain. Moreover, testing on a few applied problems certifies the theoretical deductions.