On the local convergence of efficient Newton-type solvers with frozen derivatives for nonlinear equations

Abstract

The aim of this article is to study the local convergence of a generalized $m+2$-step solver with nondecreasing order of convergence $3m+3$. Sharma and Kumar gave the order of convergence using Taylor series expansions and derivatives up to the order $3m+4$ that do not appear in the method. Hence, the applicability of it is very limited. The novelty of our article is that we use only the first derivative in our local convergence (that only appears on the proposed method). Error bounds and uniqueness results not given earlier are also provided based on q-continuity functions. We also work with Banach space instead of Euclidean space valued operators. This way the applicability of the solver is extended. Applications where the convergence criteria are tested to complete this article.