On the Local Convergence of the Third-Order Newton Iteration

Abstract

We present a local convergence analysis of the third-order Newton iteration for unconstrained optimization recently proposed by Ahmadi, Chaudhry, and Zhang. For the single-variable case, we derive the local error equation, establish cubic convergence, and obtain an explicit expression for the asymptotic error constant. A numerical example confirms the theoretical predictions. The obtained convergence order is consistent with the classical local convergence theory of Traub for one-point iterative methods.