Analysis of Henrici's Transformation for Singular Problems

Henrici's transformation is the underlying scheme that generates, by cycling, Steffensen's method for the approximation of the solution of a nonlinear equation in several variables. The aim of this paper is to analyze the asymptotic behavior of the obtained sequence (s_n^*) by applying Henrici's transformation when the initial sequence (s_n) behaves sublinearly. We extend the work done in the regular case by Sadok [17] to vector sequences in the singular case. Under suitable conditions, we show that the slowest convergence rate of (s_n^*) is to be expected in a certain subspace N of R^p. More precisely, if we write s_n^*=s_n^*,N+s_n^*,N, the orthogonal decomposition into N and N, then the convergence is linear for (s_n^*,N) but (_n^*,N) converges to the same limit faster than the initial one. In certain cases, we can have N=R^p and the convergence is linear everywhere.