Newton-like method with some remarks

Abstract

This paper presents the new convergence analysis for the Newton-like method proposed by [Xinyuan Wu, A new continuation Newton-like method and its deformation, Appl. Math. Comput. 112 (2000) 75–78.] Compared with the original version of the convergence analysis, the restriction imposed on f″( x) is removed thoroughly. In order to guarantee the quadratic convergence of the Newton-like method, it is suffices to suppose that f′( x ∗) ≠ 0 and f′( x) is local Lipschitz near x ∗, where x ∗ is a solution of nonlinear equation f( x) = 0. Moreover, some comments are given with examples for the Newton-like method, in comparison with Newton’s method. It can be concluded that the new algorithm is more feasible, effective than Newton’s method. In particularly, when it happens that, the derivative of the function f( x) at an iterate is singular or almost singular, the Newton-like method is vast superior to the classical Newton method. The numerical results of the paper strongly support the conclusion.