Abstract: | 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 (sn*) by applying Henrici's transformation when the initial sequence (sn) 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 (sn*) is to be expected in a certain subspace N of Rp. More precisely, if we write sn*=sn*,N+sn*,N, the orthogonal decomposition into N and N, then the convergence is linear for (sn*,N) but (n*,N) converges to the same limit faster than the initial one. In certain cases, we can have N=Rp and the convergence is linear everywhere. |