The application of Newton’s method in vector form for solving nonlinear scalar equations where the classical Newton method fails

In this paper we propose a strategy to obtain the solutions of a scalar equation f(x)=0f(x)=0 through solving an associated system of two equations. In this way, the solutions of the associated system lying on the identity line provide solutions of the given equation. In most cases, the associated system cannot be solved exactly. Solving this system with the Newton method for systems may result more efficient than the application of the scalar Newton method to the simpler equation. In some pathological cases in which the scalar Newton method does not provide solution, this strategy works appropriately. Some examples are given to illustrate the performance of the proposed strategy.