<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.</p></abstract>

The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing area of research in the last few years with fruitful results. Most of the studies dealt with the analysis of iterative schemes for solving non-linear equations with simple roots; however, the case involving multiple roots remains almost unexplored. The main objective of this paper was to discuss the dynamical analysis of the rational map associated with an existing class of iterative procedures for multiple roots. This study was performed for cases of double and triple multiplicities, giving as a conjecture that the wideness of the convergence regions of the multiple roots increases when the multiplicity is higher and also that this family of parametric methods includes some specially fast and stable elements with global convergence.

Stability analysis of fourth-order iterative method for finding multiple roots of non-linear equations

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

The use of complex dynamics tools in order to deepen the knowledge of qualitative behaviour of iterative methods for solving non-linear equations is a growing...

Stability analysis of fourth-order iterative method for finding multiple roots of non-linear equations

Published Online: Apr 19, 2019

Page range: 43 - 56

Received: Jan 21, 2019

Accepted: Feb 26, 2019

DOI: https://doi.org/10.2478/AMNS.2019.1.00005

Keywords
Nonlinear equations, iterative methods, multiple roots, stability, strange fixed points, free critical points

© 2019 Alicia Cordero et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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Stability analysis of fourth-order iterative method for finding multiple roots of non-linear equations

Published Online: Apr 19, 2019

Page range: 43 - 56

Received: Jan 21, 2019

Accepted: Feb 26, 2019

DOI: https://doi.org/10.2478/AMNS.2019.1.00005

KeywordsNonlinear equations, iterative methods, multiple roots, stability, strange fixed points, free critical points

© 2019 Alicia Cordero et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Keywords
Nonlinear equations, iterative methods, multiple roots, stability, strange fixed points, free critical points