Sur la convergence d'une classe de méthodes itératives de J.F. Traub

On the convergence of a class of iterative methods by J.F. Traub

Abstract

Let $X$ be a Banach space, $Y$ a normed space and $P:X\to Y$ a nonlinear operator. We study the convergence of the following method for solving the equation $P\left(x\right)=0$  $$x_{n+1}=Q\left(x_n\right)-{\left[P^{\prime }\left(x_n\right)\right]}^{-1}P(Q(x_n)),n=0,1,...,x_0\in X$$ where $Q$ is a nonlinear operator associated to the nonlinear equation $P\left(x\right)=0$. We show that if the successive approximations of $Q$ converge with order $k\ge 2$, there the above sequence converge to the solution with order $k+1$.