Nonlinear Hybrid Procedures and Xed Point Iterations

Let (x n) and (^ x n) be two vector sequences converging to a common limit. First, we shall deene nonlinear hybrid procedures which consist of constructing a new vector sequence (y n) with better convergence properties than (x n) and (^ x n). Then, this procedure is used for accelerating the convergence of a given sequence and applied to the construction of xed point methods. New methods are derived. Finally, the connection between xed point iterations and methods for the numerical integration of diierential equations is also exploited. Numerical results are given. Let (x n) be a sequence of vectors converging to some unknown limit x (sometimes, diverging sequences could also be considered). There exist many methods for accelerating the convergence of (x n). They consist of transforming (x n) into a new sequence (y n) having, for some classes of sequences, a faster convergence 9]. However, usually, such transformations, neither take into account the origin of the sequence to be accelerated nor the way its terms are obtained. In this paper, we shall discuss a sequence transformation built specially for accelerating xed point iterations in the linear and the nonlinear cases. Such a transformation will be obtained from a nonlinear extension of the hybrid procedure introduced in Section 1. Section 2 will be devoted to general results 1