New efficient bi-parametric families of iterative methods with engineering applications and their basins of attraction

Thangkhenpau, G.; Panday, Sunil; Chanu, Waikhom Henarita

doi:10.1016/j.rico.2023.100243

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…In this paper, we have presented new derivative-free three-and four-parametric with and without-memory methods for finding simple roots of nonlinear equations. The methods are based on the modifications of the derivative-free without-memory methods developed in [14]. The use of accelerating parameters in the with-memory methods has enabled us to increase the convergence order of the without-memory methods and obtain very high computational efficiency index of 7.5311 1/3 ≈ 1.9601 for the three-point and 15.5156 1/4 ≈ 1.9847 for the four-point with-memory methods.…”

Section: Discussionmentioning

confidence: 99%

“…First, let us consider the following two derivative-free families of biparametric threepoint without-memory methods [14].…”

Section: Modified Families Of Three-parametric Three-point Without-me...mentioning

confidence: 99%

“…Here, we examine the derivative-free families of biparametric four-point withoutmemory methods proposed in [14].…”

Section: Modified Families Of Four-parametric Four-point Without-memo...mentioning

confidence: 99%

“…In this paper, we introduce new derivative-free families of threeparametric three-point and four-parametric four-point with and without-memory methods for finding simple roots of nonlinear equations. The formulation of the methods is based on the derivative-free biparametric families of without-memory methods developed in [14] and by using accelerating parameters without any additional function evaluations for the with-memory methods. As a result, the orders of convergence of the with-memory methods increase from 4 to 7.5311 and 8 to 15.5156.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

et al. 2023

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In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation.

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Section: Discussionmentioning

confidence: 99%

“…First, let us consider the following two derivative-free families of biparametric threepoint without-memory methods [14].…”

Section: Modified Families Of Three-parametric Three-point Without-me...mentioning

confidence: 99%

“…Here, we examine the derivative-free families of biparametric four-point withoutmemory methods proposed in [14].…”

Section: Modified Families Of Four-parametric Four-point Without-memo...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

et al. 2023

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“…In recent years, the development of with-memory iterative methods has garnered considerable interest among researchers. For a deeper understanding, one can refer to [23][24][25][26][27][28][29][30][31][32][33][34] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

Panday,

Mittal,

Stoenoiu

et al. 2024

Mathematics

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In this article, we introduce a novel three-step iterative algorithm with memory for finding the roots of nonlinear equations. The convergence order of an established eighth-order iterative method is elevated by transforming it into a with-memory variant. The improvement in the convergence order is achieved by introducing two self-accelerating parameters, calculated using the Hermite interpolating polynomial. As a result, the R-order of convergence for the proposed bi-parametric with-memory iterative algorithm is enhanced from 8 to 10.5208. Notably, this enhancement in the convergence order is accomplished without the need for extra function evaluations. Moreover, the efficiency index of the newly proposed with-memory iterative algorithm improves from 1.5157 to 1.6011. Extensive numerical testing across various problems confirms the usefulness and superior performance of the presented algorithm relative to some well-known existing algorithms.

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A New Family of Optimal Fourth‐Order Iterative Methods for Solving Nonlinear Equations With Applications

Zein

2024

Journal of Applied Mathematics

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A new family of fourth‐order iterative methods for solving nonlinear equations is proposed using the weight function procedure. This family is optimal in the sense of the Kung–Traub conjecture, as it requires three function evaluations per iteration. Due to its flexible structure, the new family offers a variety of options, demonstrating that it includes several well‐known and recent methods as special cases. In particular, three new specific methods are designed to achieve better results compared to existing methods within the same family. Various nonlinear functions and engineering problems are used to illustrate the performance of these new specific methods, comparing them with existing ones. Furthermore, the analysis of complex dynamics and basins of attraction shows that the newly proposed methods yield the best results, with wider sets of initial points that lead to convergence.

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New efficient bi-parametric families of iterative methods with engineering applications and their basins of attraction

Cited by 3 publications

References 18 publications

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

Efficient Families of Multi-Point Iterative Methods and Their Self-Acceleration with Memory for Solving Nonlinear Equations

A New Adaptive Eleventh-Order Memory Algorithm for Solving Nonlinear Equations

A New Family of Optimal Fourth‐Order Iterative Methods for Solving Nonlinear Equations With Applications

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