Itération dirigée vers un point fixe ou un type de point fixe donné

It has long been known that roots of an equation (or a system of equations), can be obtained by first writing the system in the form xn+1 = F(xn), then deriving the recurrence system xn+1 — F(xn) ; these roots are the fixed points of the iteration, i.e. the values x0 such that x0 = F(x0). However, an iterative process may converge to or diverge from a fixed point x0, depending on the slope of F(x) at this point. Many efficient methods have been derived to promote (or accelerate) the convergence of this type of iterative process. The method described here as "directed iteration" aims to render attractive any chosen type of fixed point from a large domain of initial points. The present paper describes one-variable systems ; a subsequent paper will deal with n-variable systems. In short, we substitute for the original recurrence : [FORMULA] in which A is a positive or a negative number depending on whether the slope of F(x) at the fixed point chosen is itself positive or negative. Convergence is fastest when the value of A equals this slope ; for higher absolute values of A, convergence persists, although it is increasingly slow. More generally, one can write : [FORMULA] in which A is a caricature of the Jacobian matrix of F(x0) and I is the unitary matrix. For n-variable systems, however, it is more convenient to adopt a form closer to the Newton-Raphson recurrence : [FORMULA] in which H(xn) = F(xn) — xn and J is a caricature of the Jacobian matrix of H(x0). Our algorithm is included (as well as the Newton-Raphson and the Whit-taker algorithms) in a more general formulation given by Isaacson and Keller (1966) ; however, its usefulness in the context described here does not seem to have been considered before. As pointed out to us by C. Mira, a method developped by Giraud (1969) for two variable systems also uses ad hoc matrices which modify the attractiveness of fixed points. In the search for steady state values of systems of non-linear ordinary differential equations, directed iteration often permits convergence to a chosen steady state (or type of steady state) from a wide range of initial guesses where classical iterations fail to converge or converge from a small interval around the steady state.
Ari Richard d', Richelle Jean, Thomas René. Itération dirigée vers un point fixe ou un type de point fixe donné. In: Bulletin de la Classe des sciences, tome 73, 1987. pp. 62-83.