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 (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.
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Bellalij, M. Analysis of Henrici's Transformation for Singular Problems. Numerical Algorithms 33, 65–82 (2003). https://doi.org/10.1023/A:1025587215883
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DOI: https://doi.org/10.1023/A:1025587215883