Abstract
In this paper we analyse the dynamics of a family of rational operators coming from a fourth-order family of root-finding algorithms. We first show that it may be convenient to redefine the parameters to prevent redundancies and unboundedness of problematic parameters. After reparametrization, we observe that these rational maps belong to a more general family Oa,n,k of degree n+k operators, which includes several other families of maps obtained from other numerical methods. We study the dynamics of Oa,n,k and discuss for which parameters n and k these operators would be suitable from the numerical point of view.
Highlights
Iterative methods are the most usual tool to approximate solutions of non linear equations
It is known that the methods converge if the initial estimation is chosen suitably. The search of such initial conditions has became an important part in the study of iterative methods. To achieve this goal we analyse these methods as discrete dynamical systems
The application of iterative methods to find solutions of equations of the form f (z) = 0, where f : C → C and C denotes the Riemann sphere, gives rise to discrete dynamical systems given by the iteration of rational functions
Summary
Iterative methods are the most usual tool to approximate solutions of non linear equations. We find the analytic expressions of the regions in the parameter plane of the operator Ob where these strange fixed points are attractive and we locate these regions in the parameter space Once this initial study is done, we focus on two unwanted properties of the parameter plane of the family Ob. First, due to the fact that the coefficients of the rational map are quadratic (there are terms in b2) two different parameters b1 and b2 may lead to the same operator Ob1 = Ob2. This is an unwanted feature since may leave out of the numerical picture parameters for which relevant dynamics, such as convergence to strange attractors, take place.
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