Newton flows for real equations
The dynamical system \(N(x)=x-DG(x)^{-1}G(x),\) G: \(R^ n\to R^ n\) smooth, provided by the Newton method for \(G(x)=0\) is studied. For \(n=2\) and G rational the dynamics is understood in terms of the theory of Julia sets. Let S be the singular set of G and define the ``Julia-like'' set of N by \(J_ N=closure\{x\in R^ n| \quad N^ k(x)\in S,\quad some\quad k\}\) and the ``exploding'' set of N by \(E_ N=closure\{x\in R^ n| \quad \| N^ k(x)\| \to \infty,\quad as\quad k\to \infty \}.\) The authors give some evidence for the following conjecture (which is true for Newton's method for rational mappings of C): \(J_ N=E_ N\); and a particular model problem in \(R^ 2\) is discussed.
scientific article; zbMATH DE number 4088574