On the additional fixed points of Schröder iteration functions associated with a one-parameter family of cubic polynomials

1998 ◽  
Vol 22 (5) ◽  
pp. 629-634 ◽  
Author(s):  
V. Drakopoulos
Keyword(s):  
Fixed Points ◽  
Parameter Family ◽  
Iteration Functions ◽  
Cubic Polynomials

HOW IS THE DYNAMICS OF KÖNIG ITERATION FUNCTIONS AFFECTED BY THEIR ADDITIONAL FIXED POINTS?

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 327-334 ◽  
Author(s):  
V. DRAKOPOULOS
Keyword(s):  
Fixed Points ◽  
Rational Functions ◽  
Parameter Family ◽  
Quadratic Polynomials ◽  
Iteration Functions ◽  
Extraneous Fixed Points ◽  
Roots Of Equations ◽  
The One ◽  
Cubic Polynomials ◽  
Raphson Method

König iteration functions are a generalization of Newton–Raphson method to determine roots of equations. These higher-degree rational functions possess additional fixed points, which are generally different from the desired roots. We first prove two new results: firstly, about these extraneous fixed points and, secondly, about the Julia sets of the König functions associated with the one-parameter family of quadratic polynomials. Then, after finding all the critical points of the König functions as applied to a one-parameter family of cubic polynomials, we examine the orbits of the ones available for convergence to an attracting periodic cycle, should such a cycle exist.

scholarly journals Geometrically Constructed Family of the Simple Fixed Point Iteration Method

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 694
Author(s):  
Vinay Kanwar ◽  
Puneet Sharma ◽  
Ioannis K. Argyros ◽  
Ramandeep Behl ◽  
Christopher Argyros ◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Nonlinear Equations ◽  
Iteration Method ◽  
Parameter Family ◽  
Arbitrary Parameter ◽  
Fixed Point Iteration ◽  
Iterative Schemes ◽  
Straight Line ◽  
Predictor Corrector

This study presents a new one-parameter family of the well-known fixed point iteration method for solving nonlinear equations numerically. The proposed family is derived by implementing approximation through a straight line. The presence of an arbitrary parameter in the proposed family improves convergence characteristic of the simple fixed point iteration as it has a wider domain of convergence. Furthermore, we propose many two-step predictor–corrector iterative schemes for finding fixed points, which inherit the advantages of the proposed fixed point iterative schemes. Finally, several examples are given to further illustrate their efficiency.

scholarly journals On real fixed points of one parameter family of Function x/${(b^{x}-1)}$

2014 ◽  
Vol 46 (1) ◽  
pp. 61-65 ◽  
Author(s):  
Mohammad Sajid
Keyword(s):  
Fixed Points ◽  
Parameter Family ◽  
The Real

In the present paper, the real fixed points of one parameter family ${\cal T}=\{f_{\lambda}(x)$$=\lambda\frac{x}{b^{x}-1}\; \text{and}\;f_{\lambda}(0)=\frac{\lambda}{\ln b} : \lambda>0, x \in \mathbb{R},b>0, b\neq 1\}$ are investigated. Further, the nature of these fixed points of $f_{\lambda}(x)$ are shown for $b>0$ except $b=1$.

On the Quadratic Endomorphisms of the Plane

2004 ◽  
Vol 4 (1) ◽  
Author(s):  
F. Bofill ◽  
J.L. Garrido ◽  
F. Vilamajó ◽  
N. Romero ◽  
A. Rovella
Keyword(s):  
Fixed Points ◽  
Quadratic Forms ◽  
Degenerate Case ◽  
Parameter Family ◽  
Limit Set ◽  
Quadratic Mappings ◽  
Omega Limit Set

AbstractIn this article we establish the following result: if a nondegenerate quadratic endomorphism of the plane has no fixed points, then every point has empty omega-limit set and alpha-limit set. It is also shown that there exists a six parameter family open and dense in the space of all quadratic mappings of the plane (even those having fixed points). The degenerate case (when the quadratic forms of both components are linearly dependent), for which the theorem fails, is considered in the last section.

scholarly journals Purely Iterative Algorithms for Newton’s Maps and General Convergence

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1158
Author(s):  
Sergio Amat ◽  
Rodrigo Castro ◽  
Gerardo Honorato ◽  
Á. A. Magreñán
Keyword(s):  
Fixed Points ◽  
Large Class ◽  
Parameter Space ◽  
Critical Points ◽  
Broad Class ◽  
Iterative Algorithms ◽  
Dynamical Behaviour ◽  
Root Finding ◽  
Cubic Polynomials ◽  
General Convergence

The aim of this paper is to study the local dynamical behaviour of a broad class of purely iterative algorithms for Newton’s maps. In particular, we describe the nature and stability of fixed points and provide a type of scaling theorem. Based on those results, we apply a rigidity theorem in order to study the parameter space of cubic polynomials, for a large class of new root finding algorithms. Finally, we study the relations between critical points and the parameter space.

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