An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Zafar, Fiza; Cordero, Alicia; Torregrosa, Juan R.

doi:10.3390/math6120310

Cited by 14 publications

(6 citation statements)

References 26 publications

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“…Comparison of the proposed methods with the classical methods are numerically and graphically displayed in Table 8 and Figure 7 & Figure 8, respectively. Example 3 (beam position model [22]). Consider a mechanical issue with beam positioning that results in the following nonlinear equation:…”

Section: Some New Numerical Experimentsmentioning

confidence: 99%

Numerical Analysis of New Hybrid Algorithms for Solving Nonlinear Equations

Vivas-Cortez

Ali

Khan

et al. 2023

Axioms

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In this paper, we propose two new hybrid methods for solving nonlinear equations, utilizing the advantages of classical methods (bisection, trisection, and modified false position), i.e., bisection-modified false position (Bi-MFP) and trisection-modified false position (Tri-MFP). We implemented the proposed algorithms for several benchmark problems. We discuss the graphical analysis of these problems with respect to the number of iterations and the average CPU time.

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Section: Some New Numerical Experimentsmentioning

confidence: 99%

Numerical Analysis of New Hybrid Algorithms for Solving Nonlinear Equations

Vivas-Cortez

Ali

Khan

et al. 2023

Axioms

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“…• Now we are investigating the proposed scheme to develop optimal methods of arbitrarily high order with Newton's method, as in [26]. • Also, we are investigating to develop derivative free methods to study dynamical behavior and local convergence, as in [27,28].…”

Section: Concluding Remarks and Future Workmentioning

confidence: 99%

Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

Yan-lin

Madhu

2019

Mathematics

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The principal objective of this work is to propose a fourth, eighth and sixteenth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluations of the function and one evaluation of the first derivative; the eighth order method uses three evaluations of the function and one evaluation of the first derivative; and sixteenth order method uses four evaluations of the function and one evaluation of the first derivative. So these all the methods have satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions are compared with the existing competitors on some standard academic problems. The conjugacy maps of the presented method and other existing eighth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane. We apply the new scheme to find the optimal launch angle in a projectile motion problem and Planck’s radiation law problem as an application.

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“…Zafar et al in [32] also developed a three-point optimal iterative scheme with eighth order convergence involving free parameters as follows:…”

Section: Introductionmentioning

confidence: 99%

Generating Optimal Eighth Order Methods for Computing Multiple Roots

Kumar

Sharma

et al. 2020

Symmetry

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There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

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An Efficient Family of Optimal Eighth-Order Multiple Root Finders

Cited by 14 publications

References 26 publications

Numerical Analysis of New Hybrid Algorithms for Solving Nonlinear Equations

Numerical Analysis of New Hybrid Algorithms for Solving Nonlinear Equations

Optimal Fourth, Eighth and Sixteenth Order Methods by Using Divided Difference Techniques and Their Basins of Attraction and Its Application

Generating Optimal Eighth Order Methods for Computing Multiple Roots

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