Analysis of Henrici's transformation for singular problems (Q1410448)

In order to solve a system of nonlinear equations (1) \(F(x)=0\), where \( F:D\subset \mathbb{R}^{p}\rightarrow \mathbb{R}^{p}\) is differentiable and \(D\) is an open and convex subset of \(\mathbb{R}^{p}\), (1) is equivalently expressed in a fixed point form (2) \(x=G(x)\), where \(G\) is a certain iteration function. Then one uses the sequence of successive approximations defined by \(G\), i.e., the iterative scheme \[ x_{n+1}=G(x_{n}),n\geq 0 \tag{3} \] to obtain a fixed point of (2), that is, a solution of (1). When the iterative procedure (3) converges slowly, it is necessary to improve and accelerate its convergence. The present paper is devoted to the study of such an acceleration method, i.e., Henrici's transformation, which is the multivariable version of Aitken's \(\Delta ^{2}\) transformation.