An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points

2017 ◽  
Vol 95 (11) ◽  
pp. 2174-2211 ◽  
Author(s):  
Min Surp Rhee ◽  
Young Ik Kim ◽  
Beny Neta
Keyword(s):  
Fixed Points ◽  
Eighth Order ◽  
Extraneous Fixed Points ◽  
Newton's Methods

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 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.

scholarly journals A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics Underlying Purely Imaginary Extraneous Fixed Points

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 562 ◽  
Author(s):  
Min-Young Lee ◽  
Young Ik Kim ◽  
Beny Neta
Keyword(s):  
Fixed Points ◽  
Multiple Root ◽  
Weight Functions ◽  
Computational Results ◽  
Special Cases ◽  
Extraneous Fixed Points ◽  
Free Parameters ◽  
Future Work ◽  
Generic Family ◽  
Selection Of

A generic family of optimal sixteenth-order multiple-root finders are theoretically developed from general settings of weight functions under the known multiplicity. Special cases of rational weight functions are considered and relevant coefficient relations are derived in such a way that all the extraneous fixed points are purely imaginary. A number of schemes are constructed based on the selection of desired free parameters among the coefficient relations. Numerical and dynamical aspects on the convergence of such schemes are explored with tabulated computational results and illustrated attractor basins. Overall conclusion is drawn along with future work on a different family of optimal root-finders.

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