Newton’s Methods from a Geometric Point of View

The main problem we are going to analyze is the computation of the zeroes of an application f ∈ Ck(Rn,Rn). For many of the results in these pages we need only a small k (k = 1 or k = 2), but we are interested on the geometric behaviour of our applications, and so by convenience we always suppose k =∞. The study of the system f(x) = 0 has a long history. The case best understood is the affine situation in which the system reduces to: A · x = b, A ∈Mn×n(R), b ∈ Rn. The equation is called “linear system ” and there exists a lot of work made in its comprehension [18]. However, for the general problem: f(x) = 0 with f ∈ Ck(Rn,Rn), (1.1) unless we do suppose some restriction on the behaviour of f the problem becomes unsolvable. Being this the situation it is normal to simplify the problem by looking for points verifying weaker conditions as: x ∈ Z(f), (1.2) with Z(f) = {x ∈ Rn: exists x0 ∈ Rn such that f(x0) = 0 and ‖x−x0 ‖ < }, or more frequently: x ∈ Zf (f), (1.3) with Zf (f) = {x ∈ Rn: exists x0 ∈ Rn such that f(x0) = 0 and ‖f(x) ‖ < }