On widening region of convergence for newton' method - a numerical ODE perspective

Abstract

A heuristic analysis is given showing the connection between failures of Newton' method and solving an initial value problem of ODE'with an unstable numerical scheme. As an option to overcome the convergence difficulty encountered by Newton' method, three families of iteration methods, based on nonlinear approximations, are briefly studied. Numerical examples are presented showing the advantages of using proposed iterations in terms of having, in general, much wider regions of convergence for certain “difficult geometries.

Keywords:

• Nonlinear equation
• Newton' iteration
• Nonlinear approximation
• Global convergence
• Numerical stability of solving ODE's