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

<p>Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter <italic>A</italic> within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (<italic>A</italic> = 0), Halley’s (<italic>A</italic> = 1) and BSC (<italic>A</italic> = 2) techniques. According to the results, the one-parameter family provides the best performance for values near <italic>A</italic> = 1, which equals to the Halley’s method.</p>
</abstract>

Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter A within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (A = 0), Halley’s (A = 1) and BSC (A = 2) techniques. According to the results, the one-parameter family provides the best performance for values near A = 1, which equals to the Halley’s method.

Basins of attraction of a one-parameter family of root-finding techniques

Moroccan Journal of Pure and Applied Analysis

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

{"article-title":"Basins of attraction of a one-parameter family of root-finding techniques"}

Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting...

Basins of attraction of a one-parameter family of root-finding techniques

Published Online: Sep 29, 2023

Page range: 354 - 364

Received: Dec 10, 2022

Accepted: Sep 07, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0024

Keywords
Polynomial equations, Nonlinear equations, Iterative root-finding methods, Order of convergence, Basins of attraction

© 2023 Mário Basto et al., published by Sciendo

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

Basins of attraction of a one-parameter family of root-finding techniques

Published Online: Sep 29, 2023

Page range: 354 - 364

Received: Dec 10, 2022

Accepted: Sep 07, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0024

KeywordsPolynomial equations, Nonlinear equations, Iterative root-finding methods, Order of convergence, Basins of attraction

© 2023 Mário Basto et al., published by Sciendo

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

Keywords
Polynomial equations, Nonlinear equations, Iterative root-finding methods, Order of convergence, Basins of attraction