Superattracting Extraneous Fixed Points and n-cycles for Chebyshev’s Method on Cubic Polynomials

2020 ◽  
Vol 19 (2) ◽  
Author(s):  
José M. Gutiérrez ◽  
Juan L. Varona
Keyword(s):  
Fixed Points ◽  
Chebyshev’S Method ◽  
Extraneous Fixed Points ◽  
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 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.

scholarly journals The Optimal Order Newton’s Like Methods with Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 527
Author(s):  
Manoj Kumar Singh ◽  
Arvind K. Singh
Keyword(s):  
Fixed Points ◽  
Optimal Order ◽  
Algebraic Equations ◽  
Test Functions ◽  
Quadratic Polynomials ◽  
Nonlinear Algebraic Equations ◽  
Fractal Patterns ◽  
Extraneous Fixed Points ◽  
Order Four ◽  
Theoretical Results

In this paper, we have obtained three optimal order Newton’s like methods of order four, eight, and sixteen for solving nonlinear algebraic equations. The convergence analysis of all the optimal order methods is discussed separately. We have discussed the corresponding conjugacy maps for quadratic polynomials and also obtained the extraneous fixed points. We have considered several test functions to examine the convergence order and to explain the dynamics of our proposed methods. Theoretical results, numerical results, and fractal patterns are in support of the efficiency of the optimal order methods.

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