Complete study of local convergence and basin of attraction of sixth-order iterative method

Kasmita Devi; Prashanth Maroju; Kasmita Devi; Prashanth Maroju

doi:10.3934/math.20231080

AIMS Mathematics

2023, Volume 8, Issue 9: 21191-21207. doi: 10.3934/math.20231080

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Research article

Complete study of local convergence and basin of attraction of sixth-order iterative method

Kasmita Devi ,
Prashanth Maroju ^,

Department of Mathematics, SAS, VIT-AP University, Amaravati 522237, Andhra Pradesh, India

Received: 06 May 2023 Revised: 10 June 2023 Accepted: 15 June 2023 Published: 03 July 2023
MSC : 65H10

The local convergence analysis of the parameter based sixth-order iterative method is the primary focus of this article. This investigation was conducted based on the Fréchet derivative of the first order that satisfies the Lipschitz continuity condition. In addition, we developed a conceptual radius of convergence for these methods. Also, we discussed the solution behavior of complex polynomials with the basin of attraction. Finally, some numerical examples are provided to illustrate how the conclusions we got can be employed to determine the iterative approach's radius of convergence ball in the context of solving nonlinear equations. We compare the numerical results with our method and the existing sixth order methods proposed by Argyros et al. We observe that using our method yields significantly larger balls than those that already exist.
- local convergence,
- basin of attraction,
- nonlinear equations,
- Lipschitz continuity condition,
- Fréchet derivative
Citation: Kasmita Devi, Prashanth Maroju. Complete study of local convergence and basin of attraction of sixth-order iterative method[J]. AIMS Mathematics, 2023, 8(9): 21191-21207. doi: 10.3934/math.20231080

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Abstract

The local convergence analysis of the parameter based sixth-order iterative method is the primary focus of this article. This investigation was conducted based on the Fréchet derivative of the first order that satisfies the Lipschitz continuity condition. In addition, we developed a conceptual radius of convergence for these methods. Also, we discussed the solution behavior of complex polynomials with the basin of attraction. Finally, some numerical examples are provided to illustrate how the conclusions we got can be employed to determine the iterative approach's radius of convergence ball in the context of solving nonlinear equations. We compare the numerical results with our method and the existing sixth order methods proposed by Argyros et al. We observe that using our method yields significantly larger balls than those that already exist.

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