Reducing Chaos and Bifurcations in Newton-Type Methods

S Amat,Á A Magreñán,S Busquier

doi:10.1155/2013/726701

S Amat, Á A Magreñán + Show 1 more

Open Access

https://doi.org/10.1155/2013/726701

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Abstract

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 thebadzones 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.

Highlights

One of the most important and usual problems in numerical analysis is finding the solutions of a nonlinear equation: f (x) = 0, (1)where f : R → R is a Cr function, with r ≥ 1
We introduce a damping factor λ ∈ (0, 1] and we analyze its influence on the dynamics in the iterative method (4)
When we apply the iterative method Mλ,f to a polynomial, we may have some problems, since we obtain a rational map, say Mλ,p(x) = P(x)/Q(x), where P and Q, are polynomials, that we may suppose without common factors

Summary

Introduction

One of the most important and usual problems in numerical analysis is finding the solutions of a nonlinear equation:. Let us observe that, for a nonlinear system of m equations and m unknowns, the second Frechet derivative has m3 entries These methods are hardly used in practice. The methods consist of two (or more) steps of the Newton method having the same derivative, other main advantage of these methods relays on the fact that if we consider a system of equations only one LU decomposition is necessary in each iteration. Because of these two properties the schemes are considered a real alternative to the classical Newton method. We propose the use of damped methods in order to find good starting points for the original methods

A Damped Newton-Type Iterative Method

Quadratic Polynomials

Cubic Polynomials

Concluding Remarks