Reducing Chaos and Bifurcations in Newton-Type Methods

Abstract EN

We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three.

The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes.

The iterative schemes consist of several steps of damped Newton's method with the same derivative.

We introduce a damping factor in order to reduce the bad zones of convergence.

The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced.

Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.