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 \(P:X\rightarrow Y\) a nonlinear operator. We study the convergence of the following method for solving the equation \(P\left( x\right) =0\)  \[ x_{n+1}=Q\left( x_{n}\right) -\left[ P^{\prime}\left( x_{n}\right) \right] ^{-1}P\left( Q(x_n)\right),\ n=0,1,..., \ x_{0}\in X \] 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 \(k\geq2\), there the above sequence converge to the solution with order \(k+1\).