Two novel efficient memory-based multi-point iterative methods for solving nonlinear equations

Shubham Kumar Mittal; Sunil Panday; Lorentz Jäntschi; Liviu C. Bolunduţ; Shubham Kumar Mittal; Sunil Panday; Lorentz Jäntschi; Liviu C. Bolunduţ

doi:10.3934/math.2025250

AIMS Mathematics

2025, Volume 10, Issue 3: 5421-5443. doi: 10.3934/math.2025250

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Two novel efficient memory-based multi-point iterative methods for solving nonlinear equations

1.
Department of Mathematics, National Institute of Technology Manipur, Langol 795004, Manipur, India; shubhammehara2@gmail.com
2.
Department of Physics and Chemistry, Technical University of Cluj-Napoca, Muncii Blvd 103105, Cluj-Napoca 400641, Romania; lorentz.jantschi@chem.utcluj.ro, liviu.bolundut@chem.utcluj.ro

Received: 26 October 2024 Revised: 10 February 2025 Accepted: 14 February 2025 Published: 11 March 2025
MSC : 41A25, 65D99, 65H05

In order to solve nonlinear equations, we introduce two new three-step with-memory iterative methods in this paper. We have improved the order of convergence of a well-known optimal eighth-order iterative method by extending it into two with-memory methods using one and two self-accelerating parameters, respectively. The self-accelerating parameters that increase the convergence order are computed using the Hermite interpolating polynomial. The newly proposed uni-parametric and bi-parametric with-memory iterative methods (IM) improved the R-order of convergence of the existing eighth-order method from $ 8 $ to $ 10 $ and $ 10.7446 $, respectively. Furthermore, the efficiency index has increased from $ 1.6818 $ to $ 1.7783 $ and $ 1.8105 $, respectively. In addition, this improvement in convergence order and efficiency index can be obtained without using any extra function evaluations. Extensive numerical testing on a wide range of problems demonstrates that the proposed methods are more efficient than some well-known existing methods.
- nonlinear equation,
- roots,
- efficiency index,
- iterative method,
- with-memory methods
Citation: Shubham Kumar Mittal, Sunil Panday, Lorentz Jäntschi, Liviu C. Bolunduţ. Two novel efficient memory-based multi-point iterative methods for solving nonlinear equations[J]. AIMS Mathematics, 2025, 10(3): 5421-5443. doi: 10.3934/math.2025250

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Abstract

In order to solve nonlinear equations, we introduce two new three-step with-memory iterative methods in this paper. We have improved the order of convergence of a well-known optimal eighth-order iterative method by extending it into two with-memory methods using one and two self-accelerating parameters, respectively. The self-accelerating parameters that increase the convergence order are computed using the Hermite interpolating polynomial. The newly proposed uni-parametric and bi-parametric with-memory iterative methods (IM) improved the R-order of convergence of the existing eighth-order method from $ 8 $ to $ 10 $ and $ 10.7446 $, respectively. Furthermore, the efficiency index has increased from $ 1.6818 $ to $ 1.7783 $ and $ 1.8105 $, respectively. In addition, this improvement in convergence order and efficiency index can be obtained without using any extra function evaluations. Extensive numerical testing on a wide range of problems demonstrates that the proposed methods are more efficient than some well-known existing methods.

References

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